Skip to main content

Full text of "Rational grammar school arithmetic"

See other formats


This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project 
to make the world's books discoverable online. 

It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject 
to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books 
are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover. 

Marks, notations and other marginalia present in the original volume will appear in this file - a reminder of this book's long journey from the 
publisher to a library and finally to you. 

Usage guidelines 

Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the 
public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing this resource, we have taken steps to 
prevent abuse by commercial parties, including placing technical restrictions on automated querying. 

We also ask that you: 

+ Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for 
personal, non-commercial purposes. 

+ Refrain from automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine 
translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the 
use of public domain materials for these purposes and may be able to help. 

+ Maintain attribution The Google "watermark" you see on each file is essential for informing people about this project and helping them find 
additional materials through Google Book Search. Please do not remove it. 

+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just 
because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other 
countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of 
any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner 
anywhere in the world. Copyright infringement liability can be quite severe. 

About Google Book Search 

Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers 
discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web 



at |http : //books . google . com/ 




j ^. i -• i y _ I. i 



^-_. ilL' 



i 



1 



ISdUi^Tiiq.oG.S'^O 




HARVARD UNIVERSITY 



LIBRARY OF THE 



Department of Education 



COLLECTION OF TEXT-BOOKS 
Contributed by the Publishers 





3 2044 097 005 680 



RATIONAL GRAMMAR SCHOOL 

ARITHMETIC 



BY 

GEORGE W. MYERS, Ph. D. 

PnOFESaOR OF THE TEACHING OF MATHEMATICS AND ASTRONOMY, SCHOOL OF 
EDUCATION, THE UNIVERSITY OF CHICAGO 



SARAH C. BROOKS 

CITY NORMAL SCHOOL, BALTIMORE, MD. 



CHICAGO 

SCOTT, FORESMAN AND COMPANY 

1906 






Harvard univerei^) 
DeptofEdueattonUbrary, 

(:iftcf^h'?P:;':li.sher: 

TR/^NSFFPPED TO 

HARVARD COLLEGE LIBRARY 

^..u lb 1921 



COPYRIGHT, 1903, BY 
BCOTT, FORESMAN AND COMPANV 



ROBERT O. LAW COMPANY. 
PRINTERS AND Bl N OERS, C H IC AGO 



PREFACE 

The present book is an outgrowth of the notion that arithmetic 
as a science of pure number, and arithmetic as a school 
science, must be treated from two essentially different stand- 
points. Viewed as a finished mental product, arithmetic is an 
abstract science, taking its bearings solely from the needs, of the 
subject; but viewed as a school subject, arithmetic should be an 
abstracted science, taking its bearings mainly from the needs of 
the learner. The former calls only for logical treatment, while 
the latter calls for psychological treatment as well. In other words, 
to be of high educational value the school science of arithmetic 
must take into full account the particular stage of the pupil's 
development. The abstract stage must be approached by steps 
which begin with the learner, rise with his tinfolding powers, and 
end leaving him in possession of the outlines of the science of 
arithmetic. To break vital contact with the learner at any stage 
of the unfolding process is fatal. A controlling principle in the 
development of the various topics of this book is that any phase 
of arithmetical work, to be of value, must make an appeal to the 
life of the pupil. 

But the social and industrial factors in American communities 
enter largely into the pupil's life. This renders material drawn 
from industrial sources and from everyday affairs of high pedagog- 
ical value for arithmetic. The recent infusion of new life into the 
curricula of elementary schools through the wide introduction into 
them of nature study, manual training, and geometrical drawing 
furnishes a basis for a closer unifying of the pupil's work in arith- 
metic with his work in the other school subjects. Wide use has 
been made of all these sources of arithmetical material. 

A rational presentation of the processes and principles of arith- 
metic can be secured as well through material representing real 
conditions as through material representing artificiail conditions. 
Not only have most of the problems been draven from real sources 

m 



IV . PBEFACE ' ■ 

but a very earnest effort has also been made to have all data of 
problems absolutely correct and consistent, to the end that infer- 
ences from them may be relied upon. With so rich a store as the 
book contains, however, it is perhaps too much to hope that no 
errors remain. The authors will deem it a favor to be notified of 
any errors that may be detected. 

It is well known that the majority of the pupils of the 
alementary school never reach the high school. Even these 
pupils, whose circumstances cut them off from advanced mathe- 
matical study, have a right to claim of the elementary school some 
useful knowledge of the more powerful instruments of algebra 
and geometry. For those who will continue their studies into 
the high school it is important that the roots of the later mathe- 
matical subjects be well covered in the soil of the earlier. The 
present book seeks to meet the needs of both classes of pupils 
through the organic correlation of the elements of geometry and 
algebra with the arithmetic proper. Treated thus, the geometry 
serves to illustrate the work of arithmetic and algebra, and the 
algebra emerges from the arithmetic as generalized number. 

In particular, this text aims to accomplish four main pur- 
poses, viz. : 

(1) To present a pedagogical development of elementary mathe- 
matics, both, as a tool for use and as an elementary science; 

(2) To base this development on subject-matter representing 
real conditions ; 

(3) To open to the pupil a wide range and variety of uses for 
elementary mathematics in common affairs — to aid him to get a 
working hold of his number sense; and 

(4) To give the pupil some training in ways of attacking com- 
mon problems arithmetically and some power to analyze quanti- 
tative problems. 

It is not intended that teachers should have their classes solve 
all the problems; but rather that each teacher should select such 
topics and problems as have a particular interest for his school. 
We trust that the suggestiveness of this book as to sources of 
problems and ways of handling them in the arithmetic class will 
be appreciated by many teachers. Such teachers will doubtless 



PBBFAOB V 

prefer to work out some topics more fully than is done in the text. 
It is thought that what is given in the text will make the fuller 
working out of special topics practicable and not difficult; 

The work makes continual call for estimating magnitudes and 
for actual measurement by the pupils. Let the children be sup- 
plied with measures, foot-rules, yardsticks, meter-sticks, etc., and 
encourage their constant use. Make so regular a feature of this 
work that pupils form the habit of estimating distances, areas, 
volumes, weights, etc., always correcting their estimates by actual 
measurement. 

All models, scales, or standards of measure made by pupils 
should be carefully kept and used in the later work. 

Pupils should also be supplied with instruments with which 
the exercises in constructive geometry may be actually done. A 
cheap pair of compasses is the only necessary purchase. It is 
better that the pupil make the rest of the apparatus needed. 
If for ally reason the pupil cannot actually do the constructive 
work it should be omitted altogether. 

The Introduction which precedes the work in the formal 
operations is a departure from current text-book procedure believed 
to be worthy of special remark. It consists of fifteen pages of 
simple, practical problems, relating to matters with which the 
pupil's experiences, in school and out of school, have familiarized 
him in an indefinite way, for the right understanding of which the 
use of numbers and of the arithmetical processes is necessary. 
This chapter, by furnishing to the pupil a gradual transition from 
his vacation experiences to the rather severe study of the formal 
processes- of arithmetic, by impressing the pupil with a sense of 
the real need and purpose of such study, thus preparing his inter- 
est for it, will be recognized as a deviation from common practice 
resting upon sound pedagogical grounds. 

Especial attention is invited to this introductory chapter; to 
the chapter on measurement, §§71-81, preceding and laying the 
foundation for common fractions; to the treatment of proportion, 
§§85-97, 115-118, 190, 191; to the lists of data for original prob- 
lems, §§29, 36, 44, 132; to the work in geography, §§23, 27, 54, 
157; in farm account keeping, §35; on farm products, §62; in 



Vi PREFACE 

commerce, §28; in nature study, §§45, 122; 124, 155, 156, and 
elsewhere; in physical measurements, §§6-8, 125, 133; in paper- 
folding, §174; on the locomotive engine, §166; in constructive 
geometry, §§70, 168-180, 187; and in applied algebra, §206. 
Most teachers will approve dropping many of the time-honored 
but antiquated subjects and the curtailing of other topics 
of slight utility, and the introduction in their places of 
more timely and more real subjects. The many lists of data for 
original problems, §§29, 36, 44, 132; the treatment of longitude 
and time, §§181, 182, the descriptive work leading up to this 
topic; the extensive use of graphs to put meaning into arithmetical 
measures and to bring out the laws involved in numerical data, will 
commend themselves to teachers as useful and instructive means 
of keeping the number faculty employed on practical material. 

Another special feature is the numerous lists of problems that 
bear on the development of some important idea or law, having 
an interest on its own account. In these lists each problem is a 
step in a connected line of thought culminating in an important 
truth. This plan furnishes numerous problems, miscellaneous as 
to process, thereby requiring original mathematical thought, and 
still organically related to a central idea, thereby calling for the 
constant exercise of judgment. Examples of this may be seen in 
any section of the introduction; in the problems on geography, 
commerce, nature study, and elementary science, and also in the 
following sections: 35, 67, 68, 82, 83, 110, 115, 116, 118, 133, 
134, 136, 155, 156, 157, 160, and in practically all the matter 
from p. 264 to the end of the book. For the maturity of pupils 
of the later grades this is believed to be an important feature. 
It avoids the danger, always present with lists of promiscuous 
problems when classified under the arithmetical processes to be 
exemplified, of reducing to the mechanical what should never be 
allowed to become mechanical, viz.: the analysis of relations. 
Throughout the book the instruction is addressed to the pupil's 
understanding rather than to his memory. 

But while accomplishing this, due regard has been had to the 
necessity of sufficient drill in pure number to enable the pupil to 
obtain both a conscious recognition of processes and considerable 



PBEPAOE YU 

facility in their automatic use. This is done out of the conviction 
that the fundamental arithmetical operations should be reduced 
to the automatic stage as early as possible, consistently with a 
clear understanding of them. 

The attention of teachers is called to the section on Short 
Methods and Checking at the close of the book. After pupils 
have clearly grasped the meaning of the arithmetical processes and 
have acquired some mastery of their uses, a relief from the tedium 
of long arithmetical calculations becomes a matter of great impor- 
tance. No one can become a rapid computer without short 
methods and, considering that the problems of daily life that call 
for arithmetical treatment do not have answers with them, no one 
can be certain of his results without means of checking calcula- 
tions. Every expert accountant uses them and the more expert 
he is the more does he use them. In fact, expertness consists very 
largely in the ability to shorten calculations and to apply rapid 
checks. Training in the use of short cuts and checks should con- 
stitute a much more important part of the pupil's work in arith- 
metic than is common. Constant use of these sections should be 
made through the seventh and eighth grades. 

Definitions, processes, -rules, and even special subjects are 
worked out under the guidance of the principle: "First its 
informal, though rational, use, and afterwards conscious recogni- 
tion of its formal use." Definitions of merely technical terms are 
given when called for by the development and where the need for 
them arises. 

Unusual attention has been paid to the numerous illustrations. 
The publishers have spared no pains to make them an important 
aid to the teacher in the development of the subject. None have 
been inserted for mere adornment and none are mere pictures of 
things or relations that are perfectly obvious to the pupil. Their 
aim throughout is to secure clearness, precision, and certainty of 
thought. It is thought the book may lay rightful claim to an 
innovation in this particular. 

For convenience in reviews and for reference, a synopsis of 
definitions and a full index are given. 

The authors' acknowledgments for suggestions and ideas are 



Vlll PREFACE 

due to numerous arithmetical writers and teachers. They desire 
to express their obligations in particular to Miss Katherine M. 
Stilwell of the School of Education, University of Chicago, for 
valuable help throughout the book ; to Mr. J. B. Eussell, Super- 
intendent of Schools, Wheaton, 111., and to Miss Ada Van Stone 
Harris, Supervisor of Primary Schools, Eochester, N. Y., both of 
whom read much of the proof. They desire most heartily to 
thank Mr. Stephen Emery of Lewis Institute, whose unceasing 
diligence and pains with the proofs have wrought distinct improve- 
ments on nearly every page of the book. 

If the book shall in some measure aid in putting the teaching 
of arithmetic on a more rational basis, thereby bringing about 
results more nearly commensurate with the time and energy put 
upon the subject in the elementary schools, the authors will deem 
their efforts repaid. 

The Authors. 

Chicago, August, 1905. 



TABLE OF CONTENTS 

PAGE 

Preface iii 

Introduction 

Scale Drawing 1 

Reading Scale Drawing : 2 

Cost of Living 3 

Fencing a Farm 7 

Areas of Fields ; . , g 

Physical Measurements (a) .... ^ 11 

Physical Measurements (6) 12 

Physical Measurements (c) 13 

Wind Pressure 14 

Dairying 16 

Notation and Numeration 17 

Digits 17 

Place and Name Value of Digits 17 

Periods 18 

Decimal Notation 19 

Reading Numbers 19 

Writing Numbers 20 

Roman Notation 20 

Change of Notation 21 

Addition — Definitions 21 

Exercises 23 

Problems 24 

Distribution of Population in the United States 29 

Measurements 31 

Subtraction — Definitions 32 

Exercises 34 

Geography 37 

Commerce 38 

Numbers for Individual Work 4l 

Subtraction of Literal Numbers 42 

Multiplication — Definitions 43 

Tables 45 

Current Prices 46 

Farm Account Keeping 47 

Problems 50 

ix 



X TABLE OF CONTENTS 

PAGE! 

Multiplying by Factors. 51 

Multiplying by 10, 100, 1,000, 10.000 .' 52 

Multiplying by Numbers near 10, 100, 1,000, 10,000 52 

Multiplying by 25, 50, 12i, 75, 500, 250 52 

Multiplying when Some Digits of Multiplier are Factors of Others 53 

Checking Multiplication 53 

Multiplying by Fractional Numbers 54 

Suggestions for Problems 54 

Rainfall 56 

Algebra 59 

Division — Division and Subtraction Compared 60 

Division and Multiplication Compared 61 

Short Division 63 

Applications 64 

Long Division 65 

Exercises 67 

Larger Numbers 68 

Geography 68 

Division by Multiples of 10 70 

Other Methods of Shortening Division 72 

Tests of Divisibility 73 

Checking Division 74 

Town Block and Lots 76 

House and Furnishings 78 

House Plans 81 

Applications of Cancellation 82 

Farm Products of the United States in 1900 83 

The Thermometer 85 

Temperature Lines 86 

Examples for Practice 87 

Bills and Accounts — Exercises 88 

Accounts 88 

Problems of the Grocery Clerk 92 

Family Expense Account 93 

The Equation .*. 97 

Constructive Geometry 98 

Problems with Ruler and Compass 98 

Adding and Multiplying Lines 100 

Bisecting Lines 101 

Drawing Angles 102 

Exercises 103 

Ornamental Figures 104 

JIeasurement — Measuring Value 105 

Measuring Length and Distance 106 



TABLp OF C0NTEKT8 XI 

PAGE 

Measuring Surfaces 108 

Development of a Room 109 

The Parallelogram 110 

The Triangle Ill 

Measuring Volume (Bulk) and Capacity 112 

Measuring Weight 114 

Measuring Temperature' 116 

Measuring Time 118 

Measuring Land 119 

Plotting Observations and Measurements 120 ■ 

Heights and Weights of Boys and Girls 121 

Measuring by Hundredths 123 

Simple Interest 125 

Common Uses op Numbers 127 

Pressure of Air 127 

Passenger and Freight Trains 128 

Train Dispatcher's Report 130 

Area of Common Forms 132 

Problems 133 

Introduction to Ratio and Proportion 135 

Ratio ' 135 

Proportion 137 

Common Fractions 138 

Fractions as Ratios and as Equal Parts 138 

Reduction of Common Fractions 140 

Factors, Prime and Composite 142 

Greatest Common Divisor by Prime Factors 143 

Problems 145 

Fractions Having a Common Denominator 146 

Fractions Easily Reduced to Common Denominators 146 

Multiples 148 

Finding the Least Common Multiple 149 

Shorter Process for Three or More Numbers 150 

Definitions and Principles 151 

Addition of Fractions 153 

Subtraction of Fractions 155 

Multiplying a Fraction by a whole Number 157 

Multiplying a Mixed Number by a Whole Number 159 

Multiplying a Whole Number by a Fraction 160 

Factors may be Interchanged 160 

Multiplying a Fraction by a Fraction 162 

Multiplying a Mixed Number by a Mixed Number 163 

Dividing a Fraction by a Whole Number 165 

Dividing a Mixed Number by a Whole Number 167 



Xll TABLE OP CONTENTS 

PAOS 

Dividing any Number by a Fraction 168 

Complex Fractions 172 

Joint Effects of Forces 173 

Exercises for Practice 175 

Dividing Lines and Angles 176 

Parallel Ruler 177 

Exercises , 179 

Uses of a 30° and a 45° Triangle 180 

Drawing a Perpendicular 182 

Scale Drawings of Familiar Objects 183 

School House and Grounds 185 

Proportion 186 

Practical Applications 188 

Decimal Fractions 190 

Notation of Decimals 190 

Numeration of Decimals 191 

To Reduce a Decimal to a Common Fraction 192 

Addition of Decimals (Rain and Snowfall) 193 

Other Applications 194 

Substitution of Decimals (Nature Study) 195 

Stature and Weight of Persons 196 

Pointing the Product of Decimals 198 

Force Needed to Draw Loads on Road Wagon 199 

Division by an Integer 200 

Division by a Decimal 202 

Problems 203 

Ratio of Circumference of Circle to Diameter 204 

Original Problems 205 

Physical Measurements 206 

Specific Gravity 207 

To Reduce a Common Fraction to a Decimal 208 

The Area of a Circle 210 

Exercises for Practice 212 

Composite Denominate Numbers — Defmi ions 213 

Measures of Value 213 

Measures of Weight, . 214 

Measures of Length or Distance 214 

Measures of Surface 215 

Measures of Volume 215 

Measures of Capacity 216 

Measures of Time 217 

Measurement by Counting 217 

Exercises on Denominate Number Tables 219 

Standards of Value 219 



TABLE OF CONTENTS XIU 

PAGE 

Weight. , 4 220 

Linear Measure 221 

Surface Measure 221 

Capacity 222 

Time 223 

Counting Things 224 

General Exercises 224 

The Metric System— Historical 229 

Tables of Metric Measures 230 

Metric and U. S. Equivalents 232 

Percentage and Interest — Percentage , 235 

Algebra 238 

Gain and Loss 240 

Meteorology 243 

The Almanac 244 

Geography 245 

Commission 248 

Trade Discount 249 

Marking Goods 251 

Interest 252 

Algebra 257 

Promissory Notes 259 

Discounting Notes 260 

Partial Payments 261 

Applications to Transportation Problems — Locomotive Engine . . 264 

Laws for the Drawing of Loads 269 

Constructive Geometry — Problems 272 

To Model a 3-inch Cube 274 

To Model a Square Prism 275 

To Model a Flat Prism 276 

Comparison of Prisms \ 270 

Volume of an Oblique Prism 277 

Paper Folding 278 

Perimeters 282 

Quadrilaterals 283 

Perimeters of Miscellaneous Figures 283 

Measuring Angles and Arcs 285 

The Sum and Difference of Angles 289 

Products of Sums and Differences of Lines 293 

Maps of the World 296 

Locating and Describing Places on the Earth 297 

Longitude and Time 300 

Standard Time ; . . 303 

Mensuration— Areas, Roofing, and Brick Work 306 



XIV TABLE OF CONTENTS 

PAOB 

The Ti^apezoid 307 

Development of Octagonal Tower 309 

Land Measure 310 

Volumes 312 

Constructive Geometry 317 

The Right Triangle 320 

Squares and Square Roots 321 

Cubes and Cube Roots 326 

Similar Triangles 327 

Uses of Similar Triangles 329 

Applications of Percentage — Insurance 336 

Taxes 338 

Trade Discount 339 

Stocks and Bonds 341 

Compound Interest 343 

Use of Letters to Represent Numbers —Problems 345 

Uses of the Equation 348 

Principles for Using the Equation 349 

Problems 352 

Statements in Words and in Symbols 354 

Problems for Either Arithmetic or Algebra 356 

Formal Work 359 

Equations Containing Two Unknown Numl)era 360 

Uses of the Equation — Thermometers 363 

Applied Algebra 365 

Equivalent Readings on tlie Three Thermometers 367 

Problems 367 

Laws of Thermometer Represented Graphically 368 

Methods of Shortening and Checking Calculations Illustrations 369 

Shortening and Checking Addition 370 

Make-up Method of Subtraction 371 

Shortened Multiplication 371 

Shortened Division 373 

Shortened Square Root; Square Root by Subtraction 374 

Synopsis of Definitions 375 

General Index 386 

Answers • 



RATIOISTAL 
GRAMMAR SCHOOL ARITHMETIC 



DTTBODnCTION 



§1. Scale Drawing. oral work 

1. The scale of the drawing of the 
picture frame is 1": 4", or i. How 
high is the frame? how long? 

2. How long and how high is 
the place for the picture? Answer 
question 1 for a scale of j^f ; of 



t'j- 




t 



Bookoaae 



Q 



Mo|V|o UoUoUb 0o 



FltiUKB 1 

3. What is the scale of a drawing in 

which a line 1 in. long represents 1 ft.? in which 1 in. represents 
40 ft.? 100 ft.? 

4. In Fig. 2, which is a scale 
drawing of a schoolroom, one- 
sixteenth of an inch represents 
i ft. How long is the room? 
the teacher's desk? How long 
and how wide are the pupils' 
desks? 

5. In what direction do the 
pupils face, when seated at their 
desks? 

6. Find by measurement how 
far it is from the northwest 

corner of desk 1 to the southwest corner of desk 4; from the 
west edge of desk 2 to the east wail of i;he room. 

7. Make and answer other questions on this plan. 

8. Make a similar (like) scale drawing from measurements of 
your schoolroom and the fixed objects within it. 

9. Make a scale drawing from measurements of your school- 
house and grounds. 



Mo L b wo w p bo pU< p 



7 [ [ I I I r-r-r -T tt i i i l 1 I I I I i I I T-m 



FIOITRK 2 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



§2. Beading Scale Drawings. 

Using a foot rule marked to 8th8 and 16ths of an inch and noticing 
the scale, answer by measurement the problems on the drawing of 
figure 8. 

1. How wide is the actual block that is shown by the drawing 
MNOPl How long is the block? 



M 



A 


B 


C 


1 




D 


s 



N 



K 



m 



"4zr 



Alley 



Vegetekble 
Garden. 



^ 



m 



K 



W- 



-E 



S 
PleciLSure 

Gcxrdei\ 




Figure 3 



SCALB F/ = 100'. 



2. How wide is lot ^? B? A'? Z? 

3. How long is the vegetable garden? How wide? 

4. How wide is the pleasure garden? The southeast corner 
lot? The northeast corner lot? 



COST OF LIVING 3 

5. The alley (25 ft. wide) is taken equally from the north and 
south halves of the block. How long is lot K^ The northeast 
lot? The southeast lot? 

6. How long are the flower beds? How wide is the walk? 

7. If a man cuta a strip along the west side of lot A with a 
lawn mower that cuts a swath 1 ft. wide, how many square feet 
of the lawn does he cut in passing along the side once? Twice? 
Ten times? How many strips must he cut to mow the entire lot? 

8. How many square feet are there in lot A^ B? In the 
northeast corner lot? In the southeast comer lot? In lot K? 

9. The baseball "diamond" on the northeast lot is a 90-ft. 
square. How many square feet are there within the lines con- 
necting the bases? 

10. If a player strikes the ball and runs to third along the 
dotted lines, how far does he run? How far must a player run to 
make a "home run''? 

11. The left fielder stands at L F^nd throws the ball to the 
pitcher at P. How far does he throw? 

12. The center fielder at C F catches a ball which was struck 
at the home plate, throw:s it to third at T^ where it is caught and 
thrown to the home plate. How far has the ball gone? 

13. How much will it cost to enclose the northeast quarter of 
the block with a fence costing 25 cents a foot? 

14. Make and solve other problems on the drawing. 

15. Notice the scale of any good maps in your geography and 
find by measurement about how far it is from Chicago to New 
York; to Washington, D. C. ; to New Orleans; from New York 
to St. Louis; to Pittsburg; to Boston; to San Francisco. 

§3. Cost of Livin;. oral wore 

1. If the living expenses of a family are $70 per mo., what is 
the expense per year? 

2. If house rent is $25 per mo. and other expenses are $50, 
what part of the total expense is house rent? 

3. How many work days are there in a month? How many 
weeks in a year? 



EATIONAL GRAMMAR SCHOOL ARITHMETIC 



WRITTEN WORK 

1. Find out at home or from your neighbor how much hard 
coal is required to supply some furnace or stove 1 mo. Supposing 
hard coal costs $7.50 per T., how much will the coal cost for the 6 
winter months from November to May? 

2. If soft coal costs $3.75 a T., and If T. goes about as far as 
1 T. of hard coal, about how much would it cost to supply this fur- 
nace with soft coal for the 6 winter months? 

The prices given below are fair averages for a large city. They 
would be less in smaller places, and prices current in your com- 
munity are to be preferred to these. 



Grocer's Prices 

Apples pk. 1.35 

Apricots can . 25 

Baking powder. . . .J lb. can .20 

Bread loaf .05 

Butter lb. .28 

Cabbage lb. 2c., head .10 

Cauliflower head .15 

Celery bunch . 10 

Cheese lb. .15 

Coffee lb. .30 

Crackers lb. .10 

Eggs doz. .25 

Flour 25 lb. sack .50 

Lemons doz. .25 

Lettuce bunch .05 

Lima beans can . 10 

Milk qt. .08} 

Oatmeal 2 lb. pkg. .10 

Olives pt. .30 

Onions pk. .20 

Oranges doz. .30 

Peaches can .25 

Pears can .25 

Peas can . 121 

Pickles doz. . 10 

Potatoes pk. .25 

Prunes lb. .10 

Rice lb. .08i 

Note. — It will be assumed that 
small quantities. 



Grocer's Prices— Conhn«6d 

Rolls doz. 1.10 

Salt lb. .05 

Soap bar .05 

Starch T lb .10 

Sugar lb. .05 

Tea lb. .60 

Tomatoes can .15 



Butcher's 

Bacon 

Chicken 

Fish, fresh 

Fish, salt 

Ham 

Lard 

Lobsters, shrimps, 

Mutton 

Oysters 

Pork 

Pork tenderloin . . . 
Porterhouse steak. 

Round steak 

Salmon 

Sausage meat 

Sirloin steak 

Spare ribs 

Turkey, duck, etc. 

Veal 

rates are the same for 



Prices 

lb. 1.20 

lb. .14 

lb. .121 

lb. .10 

lb. .18 

lb. .13 

etc... lb. .25 

lb. .14 

qt. .30 

lb. .10 

. .lb. .20 

lb .18 

lb. .10 

lb. .20 

lb. .10 

lb. .16 

lb. .08i 

lb. .121 

lb. .14 

either large or 



COST OF LIVING 5 

Make such problems as : 

3. If you purchase of a grocer 2 loaves of bread, 1 lb. of butter^ 
1 lb. of coffee, and 1 pk. of potatoes, and give him $1, how much 
change should you receive, if prices are as quoted (named) in the 
table on the opposite page? 

4. If you buy a 2-lb. sirloin steak and give the butcher 50^, 
what change should you receive, prices being as in the table? 

5. Pupils may make out lists in the form of a statement and 
find the change due if the account is paid with $2, $5, or $10, 
thus : 



Bought 




• 

Paid 


Item 




Item 


1. 2 doz. rolls @ 10c. 


$0.20 


1. Cash $6.U0 


2. 1 doz. eggs 25c. 


.25 




3. 1 doz. oranges 30o. 


.30 




4. 2 bunches celery 10c. 


.20 




5. 1 lb. tea 60o. 


.60 




6. 2 cans peas 12ic. 


.25 




7. 1 lb. starch 10c. 


.10 




8. 5 bars soap 5c. 


.25 




9. 1 pkg. oatmeal 10c. 


.10 




Total cost 






Balance due in change 







6. Foot the statement above. 

7. Make out a bill of supplies such as a hotel keeper might 
purchase at the grocer's or the butcher's. 

8. Pupils may make and solve problems similar to the above, 
using local prices when convenient. 

9. The actual living expense account 
of a family for January is as shown 
in the table. If this is a fair average 
monthly expense account for the fam- 
ily, what will be its living expense for 
one year? 

10. The family pays $3000 for a 
home and is thereby saved $^6 jm 



Jan., 1902. 




Meat 


118.50 


Groceries . 


27.75 


Milk 


1.88 


Butter 


2.60 


Soft coal.. 


9.65 


Clothing . . 


15.00 





RATIONAL GRAMMAR SCHOOL ARITHMETIC 



mo. in rent. The yearly rent amounts to what part of the cost 
6t the home? 

11. If the home is in the suburbs of a city and the man must 
pay 10^ morning and evening 6 d. per wk. for car fare, how much 
does this add to his yearly expense account? 

The table given here shows the average yearly cost to one 
person for food, clothing, and household utensils from 1897 to 
1901 inclusive: 



For Year 
Ending 


Bread- 
stuffs 


Meat 


Dairy & 
Garden 


Other 
Food 


Clothing 


Metals 


Miscel- 
laneous 


Totals 


Dec.31, 1897 
Dec.31, 1898 
Dec.31, 1899 
Dec.31, 1900 
Dec.31, 1901 


$13.51 
13.82 
13.25 
14.49 
20.00 


$7.34 
7.52 
7.25 
8.41 
9.67 


$12.37 
11.46 
13.70 
15.56 
15.25 


$8.31 
9.07 
9.20 
9.50 
8.95 


$14.65 
14.15 
17.48 
16.02 
15.95 


$11.57 
11.84 
18.09 
15.81 
15.38 


$12.11 
12.54 
16.31 
15.88 
16.79 




Sums 


















Averages . . . 



















Note. — In the above table breadstuff s include wheat, corn, oats, rye, 
barley, beans, and peas ; meat includes lard and tallow ; dairy and garden 

Eroducts include vegetables, milk, butter, eggs, and fruit; the miscel- 
meous articles include a variety of things which make a part of the 
cost of living for the average family. 

12. Fill out a column like the *Hotals" column, thus finding 
the total cost of living for 1 person for each of the 5 years. 

13. What was the increase in cost of living from 1897 to 1901? 

14. Has the'cost of any of the items decreased during this 
period? How much has been the change in cost in each case? 

15. What fractional part (what part expressed as a fraction) 
of the whole cost of living for the year is the cost of breadstuflfs 
in 1897? in 1899? in 1901? 1st Arts. 4i|i = mi. 

Note. — The average of any five numbers is J of their sum. 

16. Find the average yearly cost of living for the 5 years. 

17. Find the average cost of each item (breadstuffs, meat, etc.). 

18. Fill out a column like the "totals" column, with the sums 
of the numbers in the second, third, fourth and^fth lines. What 
does each of the totals mean? 

19. Give the change, from year to year, in the total cost of 
foods for one person by finding the difference between each total 
and the one next after it. 



FENCING A FARM 



§4. Fencing a Farm. — A farm was divided into fields and seeded 
as shown in the drawing (Fig. 4). The scale of the drawing is 1 
in. to 80 rd. This means that 1 in. in the drawing represents 
80 rd. in the farm, and that all other lines longer or shorter than 
1 in. represent distances in • 

the farm proportionately 
longer or shorter than 80 rd. 

ORAL WORK 

1. How long is the corn- 
field north and south? how 
wide? 

2. How long and how 
wide is the wheatfield? the 
north oatfield? 

3. How long and how 
wide is the meadow? the 
south oatfield? 

4. The drawing of the — i p 
road is ^V i'^- wide; how 
wide is the road? 

5. How long and how wide is the farm? 

6. How long must a wire be to reach entirely round the farm? 

7. How many rods of barbed wire will be needed to inclose the 
farm with a 3- wire fence? how many miles (320 rd. = 1 mile)? 

WRITTEN WORK 

1. How many rods long is the partition (division) fence across 
the farm from east to west? what part of a mile is its length? How 
many rods of wire will make it a 4-wire fence? 1st Ans. IGO. 

2. How many rods of wire will run a 4-wire fence across the 
farm from the middle of the north side to the middle of the south 
side? Ans. 640. 

3. How many rods of wire will run a 4-wire fence between the 
wheatfield and the north oatfield? between the meadow and the 
south oatfield? 

4. How many bales of wire of 100 rd. each will be needed for 
these cross fences and division fences? AnSo 19.2. 




.Figure 4 



8 RATIONAL GKAMMAK SCHOOL ARltflMETlO 

5. How much will the wire cost for the 3-wire inclosing fence 
@ $3.50 a bale of 100 rods? Ans. $67.20. 

6. The posts for all the barbed wire fences are set 1 rd. apart. 
Hoff many posts will be needed to fence around the farm? 

7. How much will these posts cost @ 25^ apiece? 

8. How much will the wire cost @ $2.60 a bale for the two 
4-wire cross fences, and the two division fences between the wheat 
and the north oatfield and between the meadow and south oat- 
field? 

9. How much will the posts for these fences cost @ 15^ apiece? 

10. Find the total cost of wire and posts for these fences. 

11. The house and barn lot is separated from the pasture by a 

5-board fence. The boards used are 
-^^...^;' ■" '" ""■^;:::i=^;:;;;^ ^ 1 in. thick, 6 in. wide, and 12 ft. 
7' ^^E^'^^Z^^:^3!F ll! long. How many boards will reach 
L^S^^^^^^^^^^^^E ' ^"^® along the north side of the 
" ' "''^^ ~^^J^ house and barn lot? Ans. 55. 

once along the east side of the lot? 

13. How many boards will be needed for the 5-board fence 
along both sides? 

14. The posts are set 6 ft. between centers, that is, a post is 
set every 6 ft. (Fig. 5). How many posts will be needed for the 
north side? for the east side? for both? 

Ut Ans. 110; 2d Ans. 109. 

15. How much will these posts cost @ 23^ apiece? 

Lumber is sold by the hoard foot. A board foot is a board 
1 ft. long, 1 ft. wide, and 1 in., or less, thick. 

IG. How many board feet of lumber in a board 1 in. thick, 
1 ft. wide, and 5 ft. long? the same thickness and width and 10 ft. 
long? 12 ft. long? 

17. How many board feet in a board 1 in. thick, 10 ft. long, 
and 2 ft. wide? the same thickness and length and G in. wide? 9 in. 
wide? 18 in. wide? 1 in. wide? 8 in. wide? 

18. How many board feet in a board 12 ft. long, 4 in. wide, 
and 1 in. thick? 2 in. thick? 



AREAS OF FIELDS 9 

19. How many board feet in a *'two by four" scantling 
10 ft. long (2" X 4" X 10')* ? 16 feet long? 

20. How many board feet in a "four by four" scantling 10 ft. 
long? 12 ft. long? 18 ft. long? 

21. How many board feet in a fencing plank 1" x 6" x 12'? in 
5 such planks? 

22. How much will the fencing lumber of problem 12 cost @ 
$18 per M board feet? 

23. What will be the total cost of the lumber and posts for 
the fence on the north and east sides of the lot? 

24. It cost 8^ apiece to have the post holes bored, and it took 
4 days' work by 3 men @ 11.50 per day to build the fence. 
Three patent gates costing $12 apiece were put in the fence. 
Find the total cost of the lot fence, including gates. 

25. What was the total cost of all the fencing done on the 
farm? 

§5. A^reas of Fields. oral wore 

1. The whole farm contains ICO acres; how many acres are 
there in the cornfield? in the wheatfield? in the meadow? in the 
south oatfield? in the pasture? in the lot around the house and 
bam? 

2. What is the width of each of these fields in rods? 

3. If the meadow were divided by a north and south central 
line, how many rods wide would each half be? How many acres 
would there be in each half? 

4. How many square rods in a strip 1 rd. wide and 80 rd. 
long? 

5. How many square rods in an acre? 

6. How wide must a strip of land be to contain an acre if it is 
80 rd. long? 40 rd. long? i mile long? 

7. Compare the sizes of the wheatfield and the north oatfield ; 
of the cornfield and the wheatfield; of the meadow and the wheat- 
field; of the meadow and the south oatfield; of the meadow and 
the house and bam lots; of the lots and the pasture; of the 
pasture and the south oatfield. 

♦Tbe ms^Tk (') means foot or feet; (") means inch or inches. 



10 RATldXAL GRAMMAR SCHOOL ARITHMETIC 

WRITTEN WORK 

1. What would be the income per acre from the farm planted 
in corn, if the average yield were 47 bu. per acre and were sold at 
28^ per bu ? What would be the gross (total) income from the 
whole farm for this year? 

2. If the farm were rented @ $5 per acre, how much would 
remain for the tenant * from each acre? from the entire farm? 

3. What would be the owner's income from the whole farm, 
not allowing for expenses? 

4. Which is the more profitable way for the owner to rent his 
farm, @ $6 per acre cash, or for i of all the crop delivered to mar- 
ket, supposing that the whole farm is planted to corn and that a 
yield of 48 bu. per acre and a price of 30^ per bushel can be 
obtained every year? 

Note. — ^Rent of the first sort is called "cash rent," of the second sort, 
"grain rent." 

5. If this same farm will produce 20 bu. of wheat per acre 
and a price of 60^ per bushel can be obtained for it, which is the 
more profitable crop to the owner, wheat or com? how much more 
profitable? 

6. Compare the profits to the owner of a grain-rented farm 
from an oats crop of 40 bu. per acre and a price of 22^ per bushel 
with the profits of the wheat crop of problem 5 ; also with corn 
crop of problem 4. 

7. In problem 5 which would bring the larger income to the 
owner, and how much, cash rent @ $5 per A., or grain rent @ i 
delivered? 

8. Answer the same question for the oats crop mentioned in 
problem 6. 

9. A cornfield of 68 acres produced an average yield of 48 bu. 
per acre. How many bushels did the farm yield? 

10. It cost $5.85 per acre to raise the crop. At 46^ per bu. 
what was the net value of the crop? 

11. If the tenant paid cash rent @ 15.50 per acre, what was 
his net profit from the crop? How much did the owner receive? 

«The tenant is the farmer who raises the crop on another man*s farm. 



PHYSICAL MEASUREMENTS 



11 




Figure 6 



§6. Physical Measurements (a)* — A tin can, or pail, 18" high and 
10" in diameter is filled nearly full of water. A second can, or 
pail, 9" to 12" high and 9" in diameter is 
inserted, mouth downward, within the water. «^ 
A smooth hole should be punched in the p 
bottom of the smaller can. 

A rubber tube passed under the edge of B 
the inverted can and stretched over the end ^ 
of a piece of glass tubing, which is held in 
an upright position inside the inverted can 
by light cross-braces, may be used to get air. 
into the can. 

If a cork, which fits into the hole at A, is 
drawn, the inner can sinks readily in the 
water; after which the cork is inserted air- 
tight. 

A foot rule, marked with eighths or sixteenths of an inch, 
stuck with putty to the side of the inside can, permits the read- 
ings to be taken. 

Such an apparatus, when used as in the problems below, is 
called a spirometer. A more elaborate instrument is shown in 
Fig. 6. 

Common spools may be used for the pulleys that the cords pass 
over, and sand may be pressed into the cans w, w, until they just 
balance the weight of the inverted can or pail. 

1. If the inside diameter of the inverted can is 8.8 in., for each 
inch the inner can sinks into the water, there will be 60.85 cu. in. 
of water inside the can, the stopper at A being removed. How 
many cubic inches (capacity) of water will there be in the can when 
it has sunk 2 inches? 5 inches? 8 inches? 



Note. — If a pail is used, the corresponding number of cubic inches is 
found by measuring the distance in inches across the mouth of the pail, 
multiplying one-half this distance by itself, and the product by 3|. 



2. How many cubic inches of water will there be in the can 
when it is sunk in the water only 1^ inches? If inches? 2^ inches? 
2^^^ inches? 

3. Let the can now be pushed down as far as it will go and the 
stopper at A be pushed in tightly. Read the scale. When a pupil 
blows through the tube until the inside can rises 1 inch, how 



12 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

many cubic inches of air has he expelled from his lungs into the 
can? IIow many when the scale indicates that the can has riseD 
I inch? lyV inches? 1} inches? ly^ inches? If inches? 

4. Fill your lungs full and blow into the tube as long as you 
can without danger. Suppose you raise the can 2", what is the 
capacity of your lungs in cubic inches? 

5. The average lung capacity for a man 5 ft. 8 in. tall is 204 
cu. in. This average capacity may be called the normal lung 
capacity. How much does your lung capacity fall short of the 
normal value for a man? 

6. The normal lung capacity in cubic inches for a man is 3 
times his height in inches. For a woman, it is 2.6 times the 
height. Divide your lung capacity by your height in inches, and 
compare your quotient with these numbers. 

7. The chest measure of a man should be not less than half hie 
height. How much greater or less is your chest measure than it 
should be according to this law? 

§7. Physical Measurements (b). 

1. Fill out in your notebook a record like the one below, which 
is a copy from a student's notebook : 

Name, John Morrow; age, 10 yr.; height, 52 in.; weight, 61 lb. 

r on Inspiration (breathing in) S8 in. Lung capacity, 
Chest measures, ] ^^ Expiration (breathing out) ^4i in. 104 cu. in. 
* (^Mean,* 26\ in. 

Height in inches by 3 (if a boy). Result 53 X S = 156 cu. in. 
Height in inches by 2.6 (if a girl). 

Average to each inch of height, 2 cu. in., for 104. -*- 5f = ;?. 
Chest average = 26\ in. Half height = 26 in. 

2. How much do your measures exceed or fall short of the 
normal values? 

3. Do your measures show that you need chest exercise? 

4. How many in the class or room are above or below the 
normal? 



* The mean of 2 numbers is half their sum. 



PHYSICAL MEASCREMEXTS 



13 



5. Compare class or room averages* to see whether the average 
for class or room is about normal. 

6. Make the comparison of problem 5 after a course of exercises 
in physical training, and record the changes due to the exercises. 

7. Every month or two during the year find results like 
these called for in problems 2 and eo; keep your results, and show 
what changes are taking place in your measures. 

g8. Physical Measurements (c). — The following table gives the 
average height in inches and the average weight in pounds for boys 
and for girls year by year from 4 to 15 years of age. The num- 
bers are the averages of careful measures of heights and weights of 
hunflreds of boys and girls in the schools of Chicago, Boston, 
Cincinnati, and St. Louis. These averages may be called normal 
values for boys and girls of school age : 





Height 


Yearly Growth 


Weight 


Yearly Growth 


AGS 










Boys 


Girls 


Boys 


Girls 


Boys 


Girls 


Boys 


Girls 


4vr. 


39.2 in. 


39.0 in. 






87.31b. 


35.31b. 






5- 


41.6 '• 


41.4 " 






40.6" 


39.7 " 






6 " 


43.7 •' 


43.3" 






44.7" 


43.0 " 






7 " 


45.8 " 


45.5 " 






48.7 " 


47.0 " 






8 •* 


47.9 ** 


47.6 " 






58.8 " 


52.0 " 






9 ** 


49.7 " 


49.5 " 






58.7 " 


57.1 " 






10 " 


51.6 •• 


51.3 " 






64.8 " 


62.2 " 






11 " 


53.5 ** 


53.4 ** 






70.1 " 


68.1 " 






12 - 


55.2 '* 


55.8 " 






76.7 " 


77.4 " 






13 " 


57.3 " 


58.5 " 






84.9 " 


88.4 " 






14 " 


60.0" 


60.2" 






95.0 " 


98.3 " 






16 •' 


62.4 - 


61.3 " 






106.5 " 


105.0 " 







1. Subtract each number in the column of heights from the 
number next beloio it and find the growth in height for boys and 
for girls from year to year. Do the same for the weights. 

2. When is the yearly growth in height greatest for boys? 
for girls? What is the yearly growth in each case? 

3. Make and solve similar problems for weights. 

4. How much do your height and weight exceed, or fall short 
of, the normal values for your ageV 

*Tbe claims average is the sum of the measures for all pupils iu the class divided by 
tbe number of pupils. 



14 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



5. How do the averages for your room compare with the values 
in the table for the same years? 

6. How much do your height and weight exceed or fall short 
of the averages for pupils of your own age in your room? 



Wind Pressure. 



ORAL WORK 



1. Which is easier, to walk against or with the wind? 

2. If you are walking against the wind with a raised umbrella, 
why do you have to push forward harder than if you are walking 
with the wind? 

3. If you hold a board 1' wide and 10' long flatwise against the 
wind, how hard does the board push against you if the wind 
pushes against each square foot of the board with a force of 2 
pounds? 3 pounds? 5 pounds? 2^ pounds? 20 ounces? 32 ounces? 



WRITTEN WORK 

The wind is a stream of air moving along with a certain speed, 
or velocity. The greater the speed, or velocity^ the stronger does 
it push against objects it strikes 

The velocity of wind in miles per hour and the pressure per 
square foot are as given here: 



Light breeze 3| mi. .75 oz. 

Moderate breeze .. ^ ** 3.33 '* 

Fresh breeze IGJ '* 1 lb. 5 ** 

Stiff breeze 32J ** 5 *' 3 '^ 



Strong gale 56^ ml. 15 lb. 9 oz. 

Hurricane 79J " 31*' 4 ** 

Violent hurricane 97 J " 46 " 12 ** 



Note. — Use measures from your own schoolhouse or from other build- 
ings in your neighborhood in preference to the numbers in the problems. 

1. What is the total pressure in pounds on the west side of a 
house 30 ft. by 25 ft., due to a light breeze from the west? to a stiflf 
breeze? to a strong gale? 

2. What is the pressure on the side of a tall building 265 ft. 
by 80 ft., due to a hurricane blowing squarely against it? 

3. Find the wind pressure against the side of a load of hay 
25 ft. long and 12 ft. high, due to a strong gale blowing squarely 
against it. 



WIND PRESSURE 



15 



4. Find the number of pounds pressure against 
a signboard fence 100 ft. by 14 ft. , due to a strong 
gale blowing squarely against it. 

5. What is the wind pressure tending to over- 
turn a square chimney 30 ft. wide at the base, 
15 ft. wide at the top, and 175 ft. high, due to a 
.violent hurricane blowing squarely against one of 
the flat faces? 

Note. — To obtain the area of the effective surface 
against which the wind is blowing, multiply the height 
of the chimney by its breadth halfway up. This breadth 
is the half sum of the widths at the top and bottom 
(Fig. 7). 

6. Find the wind pressure on the side of a 
passenger coach 70 ft. long and 15 ft. high, due 
to a stiff breeze blowing squarely against it : 

(a) If the wind blows against the whole rec- 
tangle 15' x 70' ; 

(b) If the wind is not obstructed by the strip 
4^^' X 70', from the track to the bottom of the 
coach box and running the length of the coach. 

7. Find the wind pressure on the side of a 
train of 10 such cars as that of problem 6 (^), 
due to a strong gale, blowing squarely against it. 

8. Find the wind pressure on a diamond- 




FlGUBE 7 




FiGUBE 8 



shaped signboard, due to a violent hurricane blowing squarely 
against it, if the diamond is 12' long (AB = 12') and 6' broad 
at the middle (CD = 6'). (See Fig. 8.) 



10 



RATIOXAL GRAMMAR SCHOOL ARITHMETIC 



§10. Bairying. — The following table is an extract from the milk 
record of the University of Illinois herd of milk-cows : 

WEIGHT OF MORNING (A.M.) AND EVENING (P.M.) MILKINGS IN POUNDS 



1901 


Queen 


Beauty 


Beech- 
wood 


Mtbtlk 


Spot 


ROSE 




Not. 

1 














TOTAI. 


a.m. 
6.6 


p.m. 
6.5 


a.m. 


p.IXL 


a.ixL 


p.m. 


a.m. 


p.m. 


a.m. 
12.0 


P.J11. 
11.5 


a.in. 
11.0 


p.m. 
9.6 




11.0 


10.4 


12.5 


10.2 


7.3 


5.7 




2 


5.6 


6.0 


11.0 


8.4 


11.0 


11.0 


7.2 


6.6 


11.5 


10.5 


11.8 


8.8 




3 


6.4 


6.0 


10.8 


9.5 


11.7 


11.5 


7.3 


5.7 


12.0 


10.4 


11.2 


9.8 




4 


6.0 


6.5 


12.5 


10.2 


12.6 


13.0 


7.2 


6.0 


11.5 


11.5 


11.0 


9.8 




5 


6.0 


7.5 


11.4 


10.2 


12.0 


11.4 


6.7 


5.8 


11.6 


11.3 


11.0 


10.2 




6 


6.6 


6.8 


12.0 


10.0 


8.0 


11.6 


8.0 


5.0 


12.5 


11.8 


11.7 


10.0 




7 


7.4 


8.5 


10.8 


12.4 


12.0 


11.7 


7.0 


7.2 


12.0 


12.5 


11.6 


12.0 




8 


7.8 


7.3 


11.2 


11.6 


13.2 


12.2 


7.2 


5.7 


11.5 


11.3 


11.5 


10.1 




9 


7.5 


6.4 


11.6 


10.4 


12.0 


11.6 


6.6 


5.4 


12.0 


12.2 


10.7 


10.0 




10 


7.8 


7.2 


11.0 


10.4 


12.0 


12.0 


6.7 


4.7 


11.5 


8.5 


11.8 


10.6 




11 


6.8 


7.3 


12.2 


11.0 


13.6 


11.5 


7.7 


5.3 


9.0 


10.0 


12.3 


10.0 




12 


7.8 


7.7 


12.5 


12.0 


13.0 


11.6 


8.0 


5.5 


12.0 


12.6 


12.2 


10.6 




13 


7.6 


8.0 


13.4 


11.5 


13.5 


13.0 


7.2 


5.8 


11.1 


13.0 


11.8 


112 




14 


7.0 


8.0 


12.0 


11.8 


14.0 


12.7 


7.7 


5.7 


12.6 


12.7 


11.5 


10.3 




15 


8.8 


7.5 


13.0 


10.7 


15.0 


12.0 


7.6 


5.0 


14.0 


12.4 


12.3 


9.0 




16 


9.0 


7.0 


12.5 


11.0 


15.2 


13.7 


7.0 


5.5 


12.7 


12.3 


14.1 


8.5 




17 


8.0 


7.0 


12.5 


10.5 


14.0 


13.0 


6.1 


6.4 


10.0 


9.0 


12.0 


10.5 




18 


8.1 


7.7 


13.4 


10.9 


14.0 


11.5 


6.6 


5.0 


12.5 


11.5 


11.0 


9.1 




19 


7.0 


7.0 


11.5 


11.4 


13.6 


11.0 


6.2 


5.3 


12.2 


11.7 


11.1 


10.0 




20 


7.5 


6.5 


12.6 


10.5 


12.7 


12.0 


6.5 


4.9 


11.8 


11.4 


11.5 


9.5 




21 


7.0 


7.6 


12.0 


10.2 


13.7 


11.3 


6.1 


5.2 


13.0 


11.0 


10.6 


9.6 




22 


8.0 


7.5 


12.2 


11.9 


14.7 


11.2 


6.7 


6.0 


12.4 


10.7 


11.3 


9.5 




23 


7.5 


9.0 


12.2 


11.0 


14.4 


12.0 


7.0 


5.5 


12.8 


10.8 


11.3 


10.4 




24 


7.0 


7.1 


13.5 


10.6 


13.8 


11.5 


7.2 


5.1 


13.0 


10.7 


12.0 


10.4 




25 


7.4 


8.0 


12.5 


11.2 


14.0 


10.5 


7.4 


5.7 


12.7 


11.7 


12.0 


11.0 




26 


6.5 


8.4 


12.0 


11.8 


12.0 


12.0 


6.1 


5.8 


13.2 


11.5 


11.5 


10.5 




27 


7.5 


8.5 


13.7 


11.8 


14.0 


13.0 


6.1 


5.8 


13.0 


12.7 


11.4 


10.5 




28 


7.2 


9.2 


12.0 


12.0 


13.6 


18.8 


6.6 


49 


13.1 


12.5 


11.7 


ro.5 




29 


6.5 


9.0 


13.5 


12 6 


15.0 


13.5 


6.7 


5.0 


10.2 


12.6 


12.5 


12.6 




30 


7.2 


8.0 


11.'; 


12.7 


14.2 


12.2 


6.0 


4.3 


12.4 


11.5 


12.7 


10.3 

















1. How many pounds of milk did Queen give the first week of 
Nov., 1901, at the morning milkings? at the evening milkings? 
at both milkings? 

2. At which milking did Queen give most milk and how much 
more than at the smallest milking? 

3. Make and answer similar questions for the remaining weeks 
and for any or all of the 6 cows. 



NOTATION AKB ITOMEEATION 

§11. Digits.— 0, 1, 2, 3, 4, 5, G, 7, 8, 9. 

1. How many characters are given above? By what names do 
you know the first character? It is called zero^ or naught. 

These ten characters are called digits or figures; and by means 
of them all numbers, are expressed. 

§12. Place and Name Values of Digits. 

1. Read the following numbers: 

$30 3 30 30,000 

300 mi. .3 300 300,000 

3 yd. .03 3000 ^^ 

What is the same in all these numbers? What is different in 
the numbers? 

2. Read: 

7 feet 7 days 7 hundreds 7 millions 7 tenths 

How many feet? 7 what? How many hundreds? 7 what? 
How many millions? 7 what? In all these cases, what is the 
same? What is different? 

3. In the number CGG, what does the first G on the right 
mean? the second G? the third? 1st Ans, G ones or G units. 

4. If the value which depends on the name of a digit is called 
its name value^ what shall we call the value which depends on its 
placef 

5. How many values has a digit? What gives the first value? 
the second? Which is the permanent (fixed) value? Which is the 
changeable value? 

§13. ORAL WORK 

1. Give the place value of 2 in each of the following numbers : 
20 2 200 .2 20,000 2000 2222 

2. In 286, for what number does the 2 stand? the 8? the 6? 

17 



18 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

We may read the number 2 hundreds, 8 tens, 6 units, or two 
hundred eighty-six, meaning two hundred eighty-six units. 
Writing the names of the three places, they appear: 



I 

•a 
fl 

S3 
» 

2 8 6 



§ g fl 
» S P 



meaning 200 + 80 + 0. 

3. Read the following numbers, first as hundreds, tens, and 

units, and then as units: 

729 91G 335 494 
845 642 538 739 . 

§14, Periods. — For convenience in reading, the digits of large 
numbers are grouped into periods of three digits each. These 
periods have names depending upon their position; but the three 
places in each period are always hundreds, tens, and units.* 

The first seven periods are units, thousands, millions, billions, 
trillions, quadrillions, and quintillions. The first three periods 
are sufficient for common purposes. 



w w <n (D w 00 go 

t II I I I I 

§g? §g^ §gi Is? §g? li^ §g? 

wSp whp wSp wSp ffiSp wSp w§p 
253 462 571 324 918 645 284 

WRITTEN- WORK 

Write the following in figures : 

1. Forty-five thousand, two hundred eighty -four ; six thousand, 
nine hundred thirty-four; three hundred twenty -one thousand, one 
hundred thirteen ; five thousand, five. 

Note. — Fill all vacant places with zeros. 

*In Great Britain a notation different from that explained in $14 is used. The 
method of §14 is called the French notation and is used altogether in the United States, 
France, and Germany. The English notation groups the digits into periods of six 
figures each : 

Billions Millions Units 

865324,786593,279368 
In the English notation a mlUion millions make a billion, a million billions a 
trillion, etc. In the French notation a thousand millions make a billion, a thousand 
billions a trillion, etc. 



NOTATION AKD KtJMERATlOK 19 

2. Nine thousand, ten; ten thousand, nine; six hundred thou- 
sand, six hundred; five million, fifty thousand, three ; eighty thou- 
sand, eighty. 

§15. Decimal Notation. oral work 

1. 10 equals how many units? 1 equals what part of 10? 

2. 1 hundred equals how many tens? how many units? 1 ten 
equals what part of 1 hundred? 1 unit equals what part of 1 hun- 
dred? 

3. 1 thousand equals how many hundreds? 1 hundred equals 
what part of 1 thousand? 

4. The number denoted by each digit in 1111 is how many 
times as great as the number denoted by the digit to its right? 
The number denoted by each digit equals what part of the num- 
ber denoted by the digit to its left? 

5. How many tenths in 1 unit? 1 unit equals how many times 
1 tenth (.1)? 

6. 1 hundredth (.01) equals what part of 1 tenth? 1 tenth 
equals how many times 1 hundredth? 

7. How does the place value of the unit change from right to 
left? from left to right? 

§16. Beading Numbers. 

3,007 .6 

365,834 .27 

4,004,240 1.07 

2. Read and tell the meaning of numbers of the tables on pp. 
30, 39, or 40. 

3. Read aloud the following: 

Mean radius of the earth = 636,739,510 centimeters 

Mean distance to moon = 76,429,120 rods 

Mean radius of sun = 138,560,000 rods 

Mean distance to sun = 92,897,500 miles 

Mean distance to sun = 149,500,000,000 meters 

Light travels in one minute = 11,199,000 miles 



20 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



4. The table gives the number of vessels engaged in the trade 
of the chief countries of the world for 1901, and the number of 
tons they could carry. Eead some of the numbers: 



Country 



Greece 

Austria-Hungary 

Denmark 

The Netherlands 

Japan 

Spain 

Sweden 

Russia 

Italy 

France 

Norway 

Germany 

The United States... 
The United Kingdom 



NUMBEB 



Steamers Sailing Vessels 



150 
243 
377 
318 
511 
489 
672 
659 
381 
850 
866 

1,350 
794 

7,906 



1,078 
366 
1,200 
1,004 
2.035 
1,112 
2,328 
3,372 
1,931 
2.595 
2,913 
2,560 
4,614 
15,219 



Tonnage 



Steamers Sailing Vessels 



139,187 
294,960 
256,631 
330,132 
338,354 
460,193 
314,572 
347,875 
435,426 
547,895 
507,158 

1,561.078 
916,153 

7,944.095 



320,797 

322,897 

387,727 

451,949 

510,175 

562.068 

607,868 

850,695 

947,079 

961,259 

1,893.096 

2,106,895 

2,318,276 

10,304,338 



§17. Writing Numbers, written work 

Write in numerals the following : 

1. Four billion, two hundred thirteen million, one hundred 
twenty-two thousand, six hundred seven; forty-five billion, four 
hundred fifty million, three hundred twenty-seven thousand, one. 

2. Seven hundred eighteen billion, two hundred million ; one 
hundred ninety thousand, four hundred two; three million, one 
hundred twenty-four; fifteen thousand, nine; ninety; one hun- 
dred eighty-six billion; one hundred forty-seven million, one hun- 
dred thousand, one hundred. 

The digits, excepting zero, were first used by the Hindus, but 
they were introduced into Europe by the Arabs, and we call the 
numbers written with these digits Arabic numerals, 

§18. Roman Notation. — The Romans used letters instead of digits 
to represent numbers. This notation is still used upon monu- 
ments, and in numbering chapters and volumes of books. 
In the Roman notation, 



1= 1 


L= 50 


D= 500 


V= 5 


C = 100 


M = 1000 


X = 10 







ADDITION 



21 



When a letter of less value, as I, is placed before a letter of 
greater value, as'V, the value of the less is to be taken from that 
of the greater; thus, IV means 4 and XC means 90. 

When a letter of less value follows one of greater value, its 
value is added to that of the greater; as, VI for 6, XV for 15, 
and CX for 110. 

Repeating a letter repeats its value ; as, XX for 20, CC for 200. 

A horizontal line over a letter increases its value a thousand- 
fold; as, C meaning 100,000, D meaning 500,000. 

To interpret (change to Arabic notation) numbers written in 
the Eoman notation, always begin on the right. 

§19. Change of Notation, writtek work 

1. Change from Roman to Arabic : 

XCV CMXIX CDLXXXIV 

DXXV MLXXX MDCCLXXVI 

DCIV MCCLXX LXIX 

XOIX DLVI OCXI 

2. Change from Arabic to Eoman : 

10 65 100 572 619 1902 

30 77 500 78 584 1774 

60 80 140 50 400 1565 

49 95 46 246 2000 2590 



ADDITION 



§20. Definitions. oral work 

1. A bicyclist rides 19 mi. Monday and 10 mi. Tuesday; how 
far does he ride both days? 

2. I paid $28 for a suit of clothes and $20 for a bicycle; how 
much did I pay for both? 

3. A four-sided lot has sides of the lengths: 25 yd., 4 yd., 11 
yd., and 28 yd. ; how far is it around the lot? 

4. A man works for me 7 hr. one day, 6 hr. the next day, 7 
hr. the next, and 8 hr. the next; how many hours does he work 
for me in all? 



22 RATIONAL GBAMMAK SCHOOL ABITHXETIO 

5. A newsboy sold 18 papers Wednesday, 12 papers Thursday, 
and 16 papers Friday; how many papers did he sell in all? 

6. The number of hours the sun shone on each of the days of 
a certain week was: 8 hr., 6 hr., 3 hr., 7 hr., 5 hr., 9 hr., 2 hr. 
How many hours of sunshine were there during the week? 

7. A farmer's fields are of the following sizes : 40 acres, 80 
acres, 16 acres, and 34 acres. How many acres in all? 

8. How many dollars are $25 and $35 and $20? 

9. How many feet are 12 ft., 8 ft., 6 ft., 12 ft., and 4 feet? 

The answer to problem 1 is the same as the answer to the 
question: "What single distance is just as long as the distances 
19 mi. and 10 mi. combined?" Answering problem 2 answers the 
question: "What siugle amount equals $28 and $20 combined?" 

Combining numbers into a single number is addition. 

The result of addition is called the sum or amount. 

The numbers to be combined or added are the addends. 

Thus in the problem 

29 mi. Sum 

19 mi. and 10 mi. are the addends and 29 mi. is the sum. 

The sign of addition is +. It is read "plus." In some cases 
it may be read "and," though it is better to use the correct read- 
ing, "plus," at all times. 

The sign = when written between two numbers means that they 
are equal. It should not be read "is" or "are," but always 
"equal" or "equals." 

10. Eead and answer these questions : 

(1) 3 + 8 + 4 = ? (4) 12+ 7 + 8 = ? (7) 3 + 15 + 4 + 9 = ? 
(2)7 + 6 + 8 = ? (5) 6 + 13 + 7 = ? (8)6 + 16 + 8 + 9 = ? 
(3)9 + 8 + 5 = ? (6) 5 + 18 + 9 = ? (9)5 + 14 + 9 + 7 = ? 

11. In these questions, what numbers should stand in place of 
the question mark? 

(1) 2 8's + 5 8's = ? 8's. (5) 9 y's + 8 y\ = ? y's. 

(2) 6 9's + 7 9's = ? 9's. (6) 15 z'% + 10 z'^ = ? z'^. 

(3) 12 13's + 7 13's = ? 13's. (7) 8 rt's + 17 a's = ? a's. 
(4)* 7 a;'s + 5 a;'8 = ? re's. (8) 16 J's + 20 ^'s = ? ^'s. 

* Read 4 thus, "7 x's plus 5 aj's equal how many a's?" 



ADDITION 23 

Exercises. written work 

1. In five trips a street-car carried the following number of 
passengers: 30, 42, 45, 60, 65. Find the whole number of pas- 
sengers. 

When the sums can not be held in memory, it is convenient to 
arrange the addends in this form : 

Convenient 

FOBM 

Solution. — Writing the addends so that units, tens, . 

hundreds, etc., shall stand in the same vertical column, 30 

we add the units first; thus, 5, 10, 12 units = 1 ten and 42 

2 units. Write the 2 units in units column under the line 45 

and the 1 ten in tens column. Then, 6, 12, 16, 20, 23 tens 60 

= S hundreds and ^ tens. Write the 3 in tens column 65^ 

and the 2 in hundreds column. Then add the partial 12 

sums as shown. 23_ 

242 

2. On these trips the conductor collected $1.50, $2.10, $2.25^ 
$3.00, $3.25. The fares for the five trips amounted to what? 

3. In one week my street-car fare was 35^, 25^, 40^, 25^, 
30^, 30^. Find the whole amount. 

4. In one evening a telegraph operator sent messages contain- 
ing the following numbers of words: 2, 6, 3, 12, 6, 3, 3. What 
was the total number of words dispatched? 

5. Fred bought a note book for 10^, a lead pencil for 5^, a 
foot rule for 10^, a compass for 25^, and a bottle of ink for 10^. 
What was the amount of his bill? 

6. A train moves the following distances in five successive 
hours: 40 mi., 45 mi., 50 mi., 50 mi., 48 mi. What distance is 
traveled in the given time? 

Find the total sales in each of the following problems: 

7. Silk sales: ^ yd., 3^ yd., J yd., %\ yd., 3 yards. 

8. Linen sales: 2^ yd., 5 yd., 6 yd., 2| yd., 5 yards. 

9. Ribbon sales: 1^ yd., i yd., 3 yd., 2 yd., 12 yards. 

10. Thread sales : 25 spools white cotton, 6 spools black cot- 
ton, 3 spools black linen, 4 spools A silk, G spools twist. 

11. Handkerchief sales: 6, 12, 3, 6, 12, 6, 3, 2. How many 
dozen were sold? 



24 



EATIONAL GRAMMAB SCHOOL ARITHMETIC 



12. Kotion sales : 6 papers pins, 2 papers safety pins, 6 papers 
needles, 2 cards darning wool, 2 combs, 12 bunches tape, 4 bunches 
braid. How many articles were sold? 

13. Coffee sales: i lb., 2 lb., 2^ lb., 3 lb., J lb., 1^ lb., 5 lb., 
4 pounds. 

14. Sugar sales: 18 lb., 12 lb., 12 lb., 20 lb., 5 lb., 6 pounds. 

15. Flour sales: 50 lb., 150 lb., 26 lb., 15 lb., 25 pounds. 

16. Potato sales: 2 bu., 1^ bu., 4 bu., 6 bu., 2 bu., 1^ bu., 3 
bushels. 

17. Apple sales: 2 bu., 1 bu., 3 pk., 2 pk., 1 pk., 2 bu., 1^ 
bu., 4 bushels. (Answer in bushels.) 

18. Egg sales: 7 doz., 3 doz., G doz., 4 doz., 8 doz., 5 doz., 2 
dozen. 

19. Butter sales: 6 lb., 3 lb., 3 1b., 2i lb., H lb., 3 1b., 5 
pounds. 



§22. Problems. 





, gfS^ ► 


t 

1 


. 


1 


1 

i 

i 


Playground « 


§ Garden 

i 
\ 



FIGUBS ! 



1. A fifth grade staked out a 
rectangular plot of ground to be 
used for a playground and school 
garden. The eighth grade pupils 
fenced the plot in and ran a cross 
fence separating the parts. The 
entire plot was 126 ft. wide by 275 
ft. long (126' X 275'). How many 
feet of fence were needed? 



Solution. — Arrange the addends in convenient 
form. Beginning at the bottom of units column, 
think the sums thus, 12, 17, 23, ^8 equals 2 tens 
and 8 units. Write the 8 in units column. Add 
the 2 tens to the tens digits, thinking (not speaking) 
4, 6, 18, 15, £^ tens, equals ^ hundreds and £ tens. 
Write the 2 tens in tens place and add the hundreds 
to the hundreds in the third column, thus 8, 4, 6, 7, 
9 hundreds. Write the 9 in hundreds place, and 
since we are adding feet the sum is 928 ft. Test the correctness of the 
sum by beginning at the top of each column and adding downward. If 
the sum is the same the work is probably correct. This is called check- 
ing the work. 

2. Add 187, 892, and 478. 



Convenient Fobm 

275 ft. 

126 ft. 

275 ft. 

126 ft. 

126 ft., 

928 ft. 



Addends 



Sum 



ADDITION" 



25 



3. In a certain school there were enrolled one month : 

96 children in the kindergarten 100 in the fifth grade 

182 in the first grade 95 in the sixth grade 

143 in the second grade 83 in the seventh grade 

133 in the third grade 76 in the eighth grade 
123 in the fourth' grade 

How many pupils were in the whole school? 

4. During the same month, the numbers of cases of tardiness 
were as follows : 

Kindergarten... 10 Third grade 8 Sixth grade 12 

Eirst grade 12 . Fourth grade 5 Seventh grade.. 4 

Second grade... 4 Fifth grade 9 Eighth grade... 2 

What was the total number of cases of tardiness? 

5. The numbers of cases of absence for the same month were 
as follows, beginning with the kindergarten: 87, 60, 46, 21, 15, 
20, 15, 14, 9. Find the total number of cases. 

Teachers may take reports of their own or of other schools and 
make similar problems. 

6. Fill out for each of these cities the total number of rains 
during May and June, 1902, and the total rainfall in inches. Do 
not rewrite the numbers. 



Crrr 


Number or Rains 


Rainfall in Inches 


MAT 


June 


Total 


MAY 


June 


Total 


Chicacro 


11 
13 
14 
13 
13 
10 


15 
16 
18 
15 
16 
13 




4.46 
4.57 
3.46 
5.03 
2.83 
2.21 


6.00 
6.76 
6.16 
4.19 
7.29 
9.08 




Des Moines 


Detroit 

Kansas City 


Omaha 


St. Louis. 





The length in feet, the number of officers and men, the dis- 
placement in tons, and the indicated horse power of eight of the 
first-class battleships of the United States navy are as given in the 
table on the following page. 



26 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Name 



Length 



Men 



Displace- 
ment 



HORSB 
POWJBR 



Alabama 

Wisconsin . . . . 
Kearsarge^ . . . 

Kentucky 

Iowa 

Indiana 

Massachusetts 
Oregon 

Total . . . 



368 
368 
368 
368 
360 
348 
348 
348 



585 
585 
520 
520 
444 
465 
424 
424 



11.525 
11,525 
11,525 
11,525 
11,340 
10,288 
10,288 
10,288 



11,866 
10,000 
11,954 
12.318 
12.105 
9,738 
10,403 
11,111 



7. If these 8 vessels stood in a straight line with their ends 
touching, how far would they reach? 

8. How many officers and men are needed to man the 8 
ships? 

9. The displacement is the number of tons of water the 
vessel pushes aside as it floats. What is the combined displace- 
ment? 

10. What is the combined horse power of the engines of the 8 
ships? 

11. At the close of 1900 the numbers of teachers, schools, and 
pupils in the 6 provinces of Cuba were as follows : 



Province 



Teachers 



Schools 



Pupils 



Havana 

Puerta Principe. 

Santa Clara 

Santiago 

Pinar del Rio. . . 
Matanzas 

Total 



904 
246 
879 
645 

274 
619 



904 
246 
879 
645 
274 
619 



41.383 
9,355 
44,872 
33,983 
13,282 
29,398 



Find the totals and tell what they mean. 
12. The 6 European countries from which most immigrants 
came to the United States in 1901 were as follows: 



ADDITION 



27 



COUNTHY 


Immigrants in 1901 


Immigrants in 1900 


MALE 


Female 


Total 


MALK 


Female 


1 
Total 


Italy 

Austria Hungary 

Russia 


106,306 
78.725 
54,070 
12,894 
12,875 


29,690 
34,665 
31,187 
17,667 
10,456 




76,088 
79,978 
60,091 
16.672 
10,262 


24,047 
34,499 
31,066 
19,058 

8,388 




Ireland 


Sweden 




Total 

















Fill out all the totals and tell what they mean. 
13. The published daily circulation of a city newspaper from 
week to week is here given : 



First Week 


Second Week 


Third Week 


Fourth Week 


Fifth Week 


DAY Copies 

"1... 255, 572 
2... 303,062 
3... 323, 513 
4... 312, 724 
5... 306,009 
6... 297,918 


Day Copies 

8... 305, 725 
9... 303,489 
10... 304,991 
11... 304. 746 
12... 305, 515 
13... 299,095 


Day Copies 

15... 304.636 
16... 302, 173 
17... 302, 650 
18... 304, 255 
19... 302.942 
20... 297, 684 


Day Copies 

22... 804. 836 
23... 304,698 
24... 310.870 
25... 299, 780 
26... 801,455 
27... 298,446 


Day Copies 

29... 303,383 
30... 302.005 


Total 


Total 


Total 


Total 


Total 


Total 
for mo. 



Find the weekly totals and the total for the whole month. 

14. In the year 1891 the United States imported 94,628,119 lb. 
of coffee, which was 54,2*62,757 lb. less than was imported during 
the two previous years. How many pounds were imported during 
the two previous years? 

15. The United States imported 79,192,253 lb. of tea in 1889; 
83,494,956 lb. in 1890; and 82,395,924 lb. in 1891. How many 
pounds of tea were imported in the three years? 

16. In 1889 $27,024,551 worth of molasses was imported into 
the United States; in 1890 $31,497,243 worth; and in 1891 
$2,659,172 worth. How many dollars' worth was imported during 
the three years? 



28 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



17. Of the yearly trade within New York state, $1,650,000,000 
worth of the freight passes over the railroads; $150,000,000 over 
the canals; and $250,000,000 over Long Island Sound and the 
lakes. What is the total value of this trade? 



The Ten Largest Cities of the 
United States 

Population 

New York 3,487.202 

Chicago 1,698,575 

Philadelphia 1.293,697 

St Louis 575.236 

Boston 560,892 

Baltimore 508,957 

Cleveland 381,768 

Buffalo 352,387 

San Francisco 842,782 

Cincinnati 325,902 



The Ten Largest Foreign Cities 
Population 

London 4,433,018 

Paris 2,536,834 

Canton 2,000,000 

Berlin 1,677,304 

Vienna 1,364,548 

Tokyo 1,268,930 

St. Petersburg 1,267,023 

Pekin 1,000,000 

Moscow 988,610 

Constantinople 900,000 



18. Find the total population of the ton largest cities of the 
United States. 

Test the correctness of your addition hy adding the columns 
from top downward. 

Teachers may omit the following method of checking if thought too 
difficult. 

A very useful method of checking long problems in addition is known 
as casting out the nines. 

To cast out the nines from a number add its digits and whenever the 
sum equals or exceeds 9, drop 9 and continue adding the next digits to 
what remains, dropping 9 whenever the sum equals or exceeds 9. The 
last remainder is called the excess. 

Illustration.— Cast the 9's out of 647,255. 

Beginning on the left, 6 + 4=10; drop 9, giving the remainder 1. 
1 + 7 + 2 = 10, drop 9. 1 + 5 + 5 = 11, drop 9. The last remainder is 2 
and this is the excess of 647,255. 

To check addition by casting out nines, cast the nines out of the 
addends and the sum. Then cast out the nines from the excesses of the 
addends. If this last excess of the Excesses equals the excess of the sum, 
the work is probably correct. 



To obtain the excess of the 
excesses of the addends: 

5 + 6=11, drop 9, giving 2. 
2 + 6+5 = 13, drop 9, giving 4. 
4 + 1 = 5, the excess of the ex- 
cesses of the addends. Since this 
equals the excess of the sum, the 
work is checked. 



Illustration 


EXCESS 


8,465 
3,282 
4,497 
2.957 
7,642 


5 
6 
6 
5 
1 


26,843 


5 = 6 


5 = excess of the sum. 

5 = excess of excesses of addends. 



ADDITION 



29 



19. Find the population of the ten largest cities of Europe and 
Asia. (See table, p, 28.) 

20. Find the total population of these twenty cities. 

21. Find the population of those cities in both lists having 
more than 900,000 inhabitants. 

22. Make and solve similar problems based on the table. 

Distribntion of Population in the United States in 1900. 




FIGUBB 10 



On the map the main geographical divisions are surrounded by 
heavy full lines:—. The divisions are separated into sections 

by heavy dotted lines: . Thus the New England states are 

the eastern section of the North Atlantic division ; and New York, 
Pennsylvania, and New Jersey are the western. 

Teachers may select such problems from this list as have a 
special geographical interest to their school. 

1. From the table of numbers given on the following page, 
find the total population of the New England states for 1900. 

2. Of the North Central division. 

3. Of the Western division. 

4. Of the United States with outlying territory. 



30 



Rational orammar dcHOOL arithhetio 



5. Find the total area in square miles of the United States 
with outlying territory. 



Area and Population op 


States and Tbrb 


ITORIBS 




State or 
Tbbhitobt 


AREA 
IN SQ. 
MILBS 


Popula- 
tion 
1900 


Pupils IN 
Elem. & 

SBa 
Schools 


State or 
Tebbitort 


area 

IN SQ. 

Miles 


Popula- 
tion 
1900 


Pupils in 
£lbii. & 

Sec. 
Schools 




Me 

N. H 

Vt 

Mass. .... 

R. I 

Conn 

N.Y 

N.J 

Penn 


33,040 
9,305 
9.565 
8,315 
1.260 
4,990 

49,170 
7,815 

45.215 


694,666 

411,588 

843.641 

2,805.346 

428.666 

908.420 

7,268,894 

1.883.669 

6.802.116 


130,918 

65.193 

66.964 

474.891 

64,637 

155.228 

1,209.574 

316.055 

1,151.880 


Ky 

Tenn, ... 

Ala 

Miss 

La 

Tex. 

Okl 

Ark 

Ind.T.... 


40,400 
42,050 
62,250 
46.810 
48,720 
265,780 
39,030 
53,850 
31.400 


2.147.174 
2,020.616 
1,828,607 
1,661,270 
1,381,625 
3.048,710 

398.331 
1,811.564 

892,060 


501.893 
485.354 
376.423 
360.177 
196.169 
678,418 

314.662 
99,602 


Nortb At- 
lantic divi- 
sion 








South Cen- 
tral divi- 
sion 








Del 

Md 

D. C 

Va 

W.Va.... 

N. C 

S.C 

Ga 

Fla 


2,050 
12.210 
70 
42.460 
24.780 
52,250 
30.570 
59,475 
68.680 


184.736 
1.188.044 

278,718 
1,854,184 

958,800 
1,893,810 
1,340,316 
2,216,331 

628,542 


33.174 
229,882 

46,619 
858,825 
232,343 
400.452 
281,891 
482,678 
108,874 


Mont. . .. 

Wy 

Col 

N. M 

Ariz 

Utah 

Nev 

Idaho 

Wash. ... 

Ore 

Cal 


146,080 

97.890 

103.926 

122.680 

113.020 

84,970 

110,700 

84.800 

69,180 

96.030 

158,360 


243.329 

92.631 

639,700 

196,310 

122,931 

276,749 

42,835 

161,772 

518.103 

418.536 

1,485,058 


39,430 
14,512 

117,555 
36.736 
16,504 
73,043 
6,676 
36,669 
97,916 
89,406 

269,786 


SoTitb At- 
lantic divi- 
sion 








Western 
division .... 








Ohio 

Ind 

Ill 

Mich 

Wis 

Minn. ... 

Iowa 

Mo 

N.D 

S. D 

Neb 

Kan 


41,060 
36.360 
56.650 
58,915 
56.040 
83,365 
56,025 
69,415 
70,795 
77,650 
77.510 
82,080 


4,157,645 
2.516,462 
4,821.650 
2.420,982 
2,069,042 
1,751,394 
2.231,853 
3,106.665 
319,146 
401,570 
1,066,800 
1,470,495 


829. ICO 
564.807 
958.911 
498,665 
445.142 
899,207 
554,992 
719,817 
77 086 
96.822 
288,237 
389,588 


U. S. with- 
out outlying 
territory... 








Alaska ... 
Hawaii... 
Phil. Is.. 
Tutuila . . 
Guam . . . 
Porto \ 
Ricof 


590.884 

6,449 

114.410 

77 

150 

3,531 


(i8.592 

164,001 

6.961,339 

6,100 

9,000 

963,243 




North Cen- 
tral divi- 
sion 








Total CT. S. 
with outly- 
ing territory 

i 1 .1 









ADDITIOK 31 

6. Find the area in square miles of the North Central division. 

7. Of the South Central division. Which has the greater 
territory? 

8. What state in the Union contains the smallest number of 
square miles? the greatest? What state supports (contains) 
the greatest population? the smallest? How do the areas of 
these two states compare? 

9. Find the number of pupils attending school in the New 
England states ; in the North Central division. 

10. Find the total number of children attending school in the 
United States. 

11. Make such original problems as these: Find the total 
population in 1900 of the 13 original states; the total area of the 
13 original states; of the states east of the Mississippi river, 
etc. 

12. Check the correctness of your work in problems 2 and 7 
by adding the columns first as a whole and then adding the foot- 
ings of the separate sections and comparing the two sums. 

13. Find the population and the area of the eastern North 
Atlantic states ; of the western. 

14. Find the number of pupils attending the elementary and 
secondary schools in both the sections mentioned in problem 13. 

15. Check the work of problems 1 and 4 by casting out the nines. 

§24. Measurements. 



A square unit, or a unlt^^ square, is a square each 
of whose sides is 1 unit long. 

The area of a surface is the number of square 
units in it. 

' 1 

1. What is a square inch? a square foot? a figure u 

square yard? a square mile? a square 
rod? 

2. If a rectangle is made up of 5 
rows of 10 square units each, what 



is its area? How many tens of square 
FiGUM 19 iinits are in the rectangle? 



32 RATIONAL GRAMMAR SCHOOL ARITHMETIC 



12 



3. If a rectangle is 12 units long and 
6 units wide, how many twelves of unit 
squares are there in the area? 

4. If a rectangle 



__ is a; units long and 

Fegubb 13 . . I , 

9 units wide, how 

many a;'s of unit squares are in its area? 

Note. — Seven x's is written 7a? and is read Figubb 14 

**8«ven a;." 

5. How wide must a rectangle z inches long be to contain 6« 
sq. in.? 152; square inches? 

0. How long must a rectangle 15 in. wide be to contain 15a 
square inches? 

7. The area of one rectangle is Sz sq. in. and of another 7z sq. 
in. ; how many square inches in their sum? 

8. How many 9's are 8 9's and 7 9's? How many JI2's are 
8 I2's and 7 12's? How many 2;'s are Sx and 7x? Sz + 7z = ? 

9. Add these numbers : 



(1) 


9a 


(2) 


8a 


(3) 2G:c 


(^) 


48z 


(5) 76y 


(6) 


695 




8a 




6a 


Vtx 




98« 




49^ 




79* 


(7) 


6wi 


(8) 


(ySn 


(9) 84c 


(10) 


TM 


(11) 


imx 


(12) 


960a; 




8w 




Ibn 


76c 




39fZ 




319a; 




869a; 




9m 




98m 


37c 




79rf 




876a: 




48a; 










SUBTRACTION 










. Definitions. 




ORAL 


WORK 











§26. 

1. A man earns $180 a month and spends $140. What is the 
difference between his earnings and expenditures? 

2. One farmer owns 320 A. of land, another 80 A. Find the 
difference in size of the two farms? 

3. From a cheese weighing 43 lb., 23 lb. were sold; how many 
pounds remained? 

4. From a bin containing 72 bu. of oats, I use 32 bu. How 
•jiany bushels remain? 



SUBTRACTION 33 

5. The sum of two numbers is 48, and one of the numbers is 
28 ; what is the other? 

6. Wliat is required in the first two problems of this section? 
in the second two? in the fifth problem? 

7. What is subtraction? What is the result in subtraction 
called? 

Subtraction is the way of finding: 

(1) The difference of two numbers expressed in the same unit, 

(2) The remainder after taking a part of a number away from 
the whole number. 

The number from which we subtract is the minuend. The 
number to be subtracted is the subtrahend. The result is the dif- 
ference^ or remainder. 

Subtraction is also the way of finding either one of two addends 
when their sum and the other addend are known. The known 
sum is the minuend^ the known addend is the subtrahe7idy and the 
unknown addend is the difference^ or remainder. 

The sign of subtraction — is read "minus," and when placed 
between two numbers means that their difference is to be found. 

The minuend is always written before the sign. 

8. Tell what number the letter stands for in these problems: 
(1) 8 -5 = a; (3) l(S-l = y (5) 15-8 = ;? 
(2)17-9 = rc (4)25-y = 8 ((;)lG-;2; = 8 

9. What digit should take the place of the question mark iii 
these problems? 

(1) 7 8'8-5 8's = ?8's (4) 9:c-(;.T = ?a: 

(2) 8 12's - 3 12's = ? 12's (5) 15 x - 1 x^'i x 

(3) 4 a; - 2 a; = ? :c (G) 15 a; - ? a; =» 8 a: 

WRITTEN^ WORK 

1. A traveler has a journey of 437 mi. to make and he has 
already traveled 199 mi. of it; how much farther must he travel? 

Solution.— 437 means 400 + 30 + 7, and 199 means 100 + 90 + 9. 
We arrange the numbers conveniently, thus: 

Minuend 437 mi. = 300 mi. + 120 mi. + 17 mi. 
Subtrahend 199 mi. = 100 mi. + 90 mi. + 9 mi. 

Remainder 238 mi. = 200 mi. + 30 mi. + 8 mi. 

Beginning on the right, 9 units can not be taken from 7 units. We 
take one of the 3 tens and add it to the 7 units, giving 17 units. Then 17 
units — 9 units = 8 units. Write a' in units column. Passing to tens 



34 RATIONAL OIlAMMAR SCHOOL AElTHMMlO 

column, we can not take 9 tens from the 2 tens remaining, so we take one 
of the 4 hundreds and add it to 2 tens, making 12 tens. Then 12 tens — 9 
tens = 3 tens. Write 3 in tens column. Finally, 3 hundreds — 1 hundred 
= ^ hundreds. Write the fJ in the hundreds column. The Vemainder 
is 238 miles. 

2. How can yoa prove 238 to be the correct remainder or 
difference in problem 1? 

3. The minuend is 1284, and the subtrahend is 876. What is 
the difference, or remainder? Prove. 

4. When the water was dried out of 38 oz. of natural soil, the 
dry soil weighed 29 oz. Make a problem from these facts, solve 
it, and prove your solution correct. 

0. A subtrahend is 99, and the difference is 338. What is 
the minuend? IIow is it found? 

6. From 72 hr. subtract 1 da. and 4 hours. 

Note. — A correct answer to this problem would be 72 hr. — 1 da. and 4 
hr. But the answer is to be expressed in a single unit, or name. To do 
this, first change 1 da. and 4 hr. to hours. 1 da. = 24 hr. ; 24 hr. + 4 hr. 
= 28 hr. 72 hr. — 28 hr. = 44 hours. 

7. From 30 ft. take 5 yd. What must first be done to obtain 
the result in a single unit? Make complete statements. 

8. 20 qt. - 2 pk. = x qt. What is the first thing to be done? 
Find the value of x. 

9. From J yd. take | yd. What is the unit of J yd.? of f yd.? 

What then must be done before subtracting? Why? 

Note. — Only numbers expressed in the same unit can be subtracted 
if the result is to be expressed in a single unit. 

10. Write the last problem, using the sign of subtraction. 

§26. Exercises. 

1. From a barrel of vinegar containing 31| gal., suppose the 
following quantities to be removed, and give successive remainders: 
G gal. ; 2i gal. ; 5 gal. ; 3 gal. ; 2^^ gal. ; 2^ gallons. 

2. From a wagon box containing 42 bu. of potatoes, a farmer 
sold the following quantities: 2 bu,, 4 bu., 3 bu., 2 bu., 5 bu., 
6 bu., 9 bu., 4 bu. How many bushels remained? 

3. On March 31 the gas meter read 30,000 cu. ft., and on 
April 30, 45,000 cu. ft. How many cubic feet of gas had been 
used durii^g the month? 



StTBTRACTIOK 35 

4. A horse and his rider weigh 1375 lb. The man weighs 
160 lb. What is the weight of the horse? 

5. Find and read the differences in these exercises: 

(1) (2) (3) 

$9274.75 $4246.38 1,024,637 yd. 

1846.19 ' 1571.16 725,448 " 



(2) 
$4246.28 
1571.16 


(2) 
$3472.06 
2189.24 



6. Find the differences : 

(1) (2) (3) 

6,420,014 rd: $3472.06 1228f 

1,382,741 '' 2189.24 936^ 

7. During one week a merchant's transactions (or business 
dealings) at his bank were as follows: deposits (moneys put in), 
$217.40, $343.27, $290.00, $365.18, $380.24, $415.17; with- 
drawals (moneys taken out), $200, $320.25, $195.75, $286, $240.16, 
$395.25. Which item was the greater at the close of the week, 
and how much? 

8. During one week of Xovember, 1901, 63,188 cattle were 
received at the Chicago stock yards; during the corresponding 
week of 1899, 32,722 were received. What was the increase? 

9. Great Salt Lake is 4200 ft. above sea level, and Lake Su- 
perior 602 ft. X is the difference in elevation. Find value of x. 

10. Cuba contains 45,884 sq. mi., and the Philippines, 114,410 
sq. mi. Find the difference in areas. 

For the numbers for problems 11-17, see the tables^ p. 36. 

11. Find, the total receipts at Chicago for the week Xov. 
25-30, 1901, of cattle; of calves; of hogs; of sheep. 

12. Find the total shipments. 

13. Find the difference between receipts and shipments. 

14. Compare the receipts with those of the previous week; 
with those of the corresponding week of 1900; of 1890. 

15. Compare shipments in the same manner. 

10. Find total receipts for one week in the four markets, Chi- 
cago, Kansas City, Omaha, and St. Louis, of cattle; of hogs; of 
sheep. 

17. Compare Chicago receipts with those of tjie other cities. 



36 



EATIONAL GRAMMAR SCHOOL ARITHMETIC 



A Week's Receipts and Shipments op Live Stock. 
Chicago, Nov. 30, 1901 



RECEIPTS 




Cattle 


Calves 


Hogs 


Sheep 


Monday, Nov. 25 


16.078' 

6.941 

17,583 

5.857 
800 


537 
735 

481 

162 
40 


39,630 
41,654 
51,275 

44.090 
25,000 


26,170 i 

18,297 

12,393 

12.462 
2,000 


Tuesday, " 26 


Wednesdav. ** 27 


Thursday. ** 28 (Holiday) . . 
Friday, '* 29 


Saturday, " 30 




Total 


63.188 
53.965 
32,722 


3,814 
1.822 
1,335 


273.426 
191,104 
159,724 


104.528 
55,046 
61,718 


Previous week 

Corresponding week 1900 

Corresponding week 1899 


shipments 




Cattlb 


Calves 


Hogs 


Sheep 


Mondav Nov 25 


2.578 
1,904 
5,647 

1,875 
400 


2 

1 
16 

205 
30 


7,543 
1.495 
5,712 

4,308 
5,000 


2,a52 
4,518 
4,501 

2,127 
1,000 


Tuesdav. ** 26 


Wednesday^ ** 27 


Thursday, ** 28 (Holiday) . . 
Friday, " 29 


Saturdav. *' 30 




Total 


18,699 

18.438 

9,556 


324 
352 

210 


29,486 
23.675 
16,637 


24,429 

14,832 

3,182 


Previous week 


Corresponding week 1900 

Corresponding week 1899 



Live Stock Receipts for One Week at Four Markets 



Cattle 



Hdbs 



Sheep 



Chicago 

Kansas City 

Omaha 

St. Louis 

Total 

Previous week 

Corresponding week 1900 
Corresponding week 1899 
Corresponding week 1898 
Corresponding week 1897 



47.300 
29,300 
16,800 
10,700 



201,600 
89,000 
65,200 
38,100 



71,300 

14,900 

12,800 

5,800 



144.600 
111.500 
91.700 
128.400 
143,000 



481,700 
339,800 
206.100 
365.500 
374,100 



173,600 
85,000 
81,300 
95.600 

105,600 



SUBTRACTION" . . ^'^ 

18. Compare the total receipts of the week with those 
of the previous week; of the corresponding week of 1900; 1899; 
.1898; 1897. 

§27v Geography. 

1. Lake Titicaca is 12,645 ft. above the level of the sea, and 
Lake Superior 602 ft. Find the difference in elevation. 

2. Mt. Everest is 29,002 ft. above sea level, and Mt. Blanc 
15,744 ft. What is the difference in altitude? 

3. The Mississippi basin has an area of 1,250,000 sq. mi. ; 
the Amazon, of 2,500,000 sq. mi. How much greater is the basin 
of the Amazon than that of the Mississippi? 

4. The area of the Philippines is 114,410 sq. mi. ; that of 
California, 158,360 sq. mi. The state is how much larger than 
the islands? 

5. Compare the same islands with Texas; with Illinois; with 
Rhode Island. (See table, p. 30.) 

6. Of a population of 6,302,115 in Pennsylvania, 1,151,880 
are pupils in elementary and secondary schools. How ma-ny per- 
sons are not attending those schools? 

7. In a certain year the Hawaiian Islands produced 221,694 
T. of sugar, while Cuba produced 880,372 T. How much more 
did Cuba produce than the Hawaiian Islands? 

8. In 1890, 25,403 manufacturing establishments in Xew 
York City paid for wages $230,102,167; for materials, $366,422,- 
722; for miscellaneous expenses, $59,991,710. The value of the 
products was $777,222,721. AVhat was the total profit for all the 
establishments? ' 

9. The battle of Lexington was fought in 1775, and the 
battle of Santiago in 1898. How many years elapsed between the 
two battles? 

For the numbers for problems 10-19, see table, p. 30. 

10. What is the difference of the areas of the largest and the 
smallest states? 

11. How many more persons are in the state with the largest 
population than in the one with the smallest? 



38 RATIONAL ORAMMAB SCHOOL ABITHKETIG 

12. How many more children attend school in one of these 
two states than in the other? 

13. Make and solve other problems, comparing the areas, 
total populations, and school populations of the different states. 

14. Which is the larger, and by how much, the North Atlantic 
division or the South Atlantic division? the North Central or the 
South Central? the South Central or the Western? 

15. Compare the total populations of these groups. 

16. How many {arsons are not attending elementary or second- 
ary schools in Maine? in Mississippi? How do these two results 
compare? 

17. Find how many persons are not attending these schools in 
your own state ; in bordering states. 

18. How many more persons are not attending these schools in 
New York than in Illinois? How does this result compare with 
the difference in the total populations of these states? 

19. Make and solve other problems of interest to your school. 

20. Find, by comparing the populations for 1890 (p. 39) with 
those for 1900 (p. 30), the increase of population for the ten years 
in each state of the North Atlantic division ; in the whole division. 

21. Find the increase of population in your own state from 1890 
to 1900. What is the average yearly increase? 

22. How many years since Illinois was admitted as a state? 
since your native state was admitted into the Union? 

23. Make similar problems from the table opposite and the 
table on p. 30, solve your problems, and check your work. 

24. Each principal division of states is separated, into two or 
more sections by heavy dotted lines on the map, p. 29. Find 
how much greater, or less, the growth of population since 1890 
has been in your section than in the other section, or sections of 
the same division. 

§28. Commerce. 

1, Find the total export trade of the United States with the 
twelve countries given in the table on p. 40, in 1891 and in 1901. 

2. Find the increase, or decrease, for each country, and make 
a list of these results. Whenever the difference is an increase 
write + before it. When it is a decrease write - before it. Make 
the subtractions without rewriting the numbers, recording results. 



fitJBTRACTIOK 



3d 



Dates of Admission and Populations for 1890 of States and 
Territories Geographically Grouped 





Date op 


POPULA- 




Date of 


Popula- 




Admission 


TION 1890 




Admission 


tion lim 


Me 


Mar. 15, 1820 


661.086 


Ky 


June 1, 1792 


1,848,635 


N. H 


June 21, 1788 


376.530 


Tenn 


June 1, 1796 


1.767.518 


Vt 


Mar. 4, 1791 


332.422 


Ala 


Dec. 14, 1819 


1,513,017 


Mass 


Feb. 7, 1788 


2,238.043 


Miss 


Dec. 10, 1817 


1.289,600 


R. I 


May 20, 1790 


345.506 


La 


April 30, 1812 


1,118,587 


Ck)nn. . . . 


Jan. 9, 1788 


746,258 


Tex 


Dec. 29. 1845 


2,235,528 


N. Y 


July 26. 1788 


5,997,853 


Okl 


May 2. 1890 


61,834 


N.J 


Dec. 18. 1787 


1,444,933 


Ark 


June 15, 1836 


1,128,179 


Penn. . . . 


Dec. 12, 1787 


5,258,014 


Ind. T. .. 


June 30, 1834 


180,182 


North . 


Atlantic divi- 




South Central divi- 




sion . 






sion . 










Del 


Dec. 7, 1787 


168,493 


Mont. . . . 


Nov. 8, 1889 


132,159 


Md 


April 28. 1788 


1,042,390 


Wy 

Col 


July 10. 1890 


60,705 


D. C 


Mar. 30, 1791 


230.392 


Aug. 1, 1870 


412,198 


Va 


June 25, 1788 


1.656,980 


N. M 


Sept. 9, 1850 


153.593 


W. Va. . . 


June 20, 1863 


762,794 


Ariz 


Feb. 24, 1863 


59.620 


N. C 


Nov. 21, 1789 


1,617,947 


Utah 


Jan. \ 1896 


207,905 


S. C 


Mar. 23. 1788 


1,151,149 


Nev 


Oct. 31, 1864 


45.761 


Ga 


Jan. 2. 1788 


1.837,353 


Idaho 


July 3. 1890 


84,885 


Fla 


Mar. 3, 1845 


391.422 


Wash. . . . 


Nov. 11. 1889 


349,390 








Ore 

Cal 


Feb. 14. 1859 
Sept. 9, 1850 


313.767 
1,208,130 








South J 
sion . 


Atlantic divi- 










Western division 






Ohio 


Feb. 19, 1803 


3,672,316 






Ind 


Dec. 11. 1816 


2.192.404 


U. S. without out- 




Ill 


Dec. 3, 1818 


3.826,351 


lying territory. . . • . 




Mich.. .. 
Wis 


Jan. 16, 1837 
May 29, 1848 


2.093,880 
1,686,880 












Minn. . . . 


May 11.1858 


1.301,826 


Alaska . . 


July 27. 1868 


32,052 


Iowa 


Dec. 28, 1846 


1,911,896 


Hawaii. . 


April 30, 1900 


89,990 


Mo 


Aug. 10, 1821 


2,679,184 


Phil. Is. . 






N. D 


Nov. ^, 1889 


182,719 


Tutuila.. 






S. D 


Nov. 2. 1889 


328.808 


Guam . . . 






Neb 


Mar. 1, 1867 


1,058,910 


Porto ) 
Rico J 






Kan 


Jan. 29, 1861 


1,427,096 






North i 


Central divi- 




Total U. S. with out- 




sion . 






Ivine: fArrifnrv 






'J *"6 


' 



40 



RATIONAL ORAltMAft SCHOOL ARltMMEtlC 



United States Export Trade 



Country 



United KiDgdom . . 

(rerniany 

Canada 

Netherlands 

Mexico 

Italy 

British Australasia 

British Africa 

Japan 

Brazil 

Argentina 

Russia 

Total 



1901 



$598,766,799 

184.678.723 

107,496.522 

85,643.804 

36,771,568 

84.046,201 

30,569,814 

24,994.766 

21.162,477 

11,136.101 

11,117.521 

6,504,867 



1891 



9482,295, 796 

90,826,332 

41.686,882 

81,261,766 

15,371,870 

14,447,004 

13,564,931 

3.511,668 

3.839,384 

15.064,346 

1,909,788 

6,400.857 



Ten-Ybab 
Difference 



Values op Manufactured and Industrial Products 



Manufactured Articles 


19(tt 


1897 


Difference 


Agricultural implements 

Books mans, etc 


82,075,609 
988,195 
913,513 
198,438 
385,086 
1,634,642 
98,476 
735,165 
182,710 
894,330 


« 248,466 

470,358 

80.065 

31,583 

1,499,769 

988,661 

839,563 

877.549 

69.756 

1,222,708 




Carria&res and cars 


Copper ingots 


Cotton cloths 


Cotton manufactures, other 

Cycles, and parts of 


Builders' hardware 


Sewincr machines 


Other machinery 




Total 










Other Articles 


190S 


1897 


Difference 


Corn 


»1.468,390 

3,769,577 

638,361 

5,473,177 

4.509,205 

1,345,260 

667,164 

261,688 

240,978 

557,827 

218,995 


»1, 770.581 

2,548,778 

2,415.519 

6,987,856 

2,626,679 

566.584 

195,584 

47,069 

208.195 

365,419 

188,116 




Wheat 


Wheat flour 


Coal 


Cotton 


Fruits and nuts 


Furs and fur skins 


Cotton-seed oil 


Beef salted or oickled 


Bacon 


Hams 




Total 











SUBTRACTION 



41 



3. The columns on the opposite page show the values of the 
different kinds of goods exported by the United States to Canada 
during the nine months ending March, 1902, and March, 1897, 
respectively. Make a list of diflPerences, marking the difference + 
whenever it denotes an increase, and - when it denotes a decrease. 

§29. Nnmberg for Individual Work. 

School Statistics for the Thirty Largest Cities of United States 



City- 


Popula- 
tion. 
Census 

1900 


POPUIiA- 
TION. 

Census 

1890 


School 
Enroll- 
ment. 
1900 


NUMBKK 
TEACH- 
ERS. 

1900 


School 

EXPENDI- 

TUKKS. 

1900 


New York 


3,437,202 

1,698,575 

1,293,697 

575,236 

560,892 

508,957 

381,768 
352,387 
342,782 
325,902 
321,616 
287.104 
285,704 
285,315 

246,070 
218,196 
206,433 
204,731 
202,718 
175,597 
169.164 
163,752 
163,065 
162,608 

133,859 
131,822 
129,896 
125,560 
118.421 
108,374 


2,492,591 

1,099,850 

1,046,964 

451,770 

448,477 

434,439 

261,353 
255.664 

298,997 
296,908 
238,617 
242,039 
205,876 
204,468 

181,830 
188,932 
163,003 
161,129 
164,738 
132,146 
105,436 
132.716 
133.156 
133,896 

106,713 
81,434 

105,287 
88.150 
84.655 
88,143 


559,218 

262,738 

151,455 

82,712 

91,796 

65,600 

59,635 
56,000 
48,517 
44,285 
50,000 
31,547 
40,303 
87,000 

41,870 
40.069 
32,174 
27.626 
38,591 
23.485 
27.334 
28.280 
26,000 
24.896 

27,181 
21.467 
20.104 
18,855 
19,600 
21,090 


12,212 
5,951 
3,591 
1,751 
2,018 
1,600 

1,303 
1,300 
1,017 

993 
1,000 

782 
' 966 

900 

851 
1,043 
686 
650 
892 
682 
650 
700 
610 
692 

530 
455 
377 
502 
574 
485 


$21,040,810 
7.929,496 
4,677,860 
1.526.140 
3,664,298 
1,279,936 

1,933.965 
1,408,000 
1,152,631 
1,064.047 
1,757.381 

455.073 
1,251.825 

733,510 

1,218,660 

634.153 
555,811 
841,000 
682,000 
729,106 
524,065 
672,350 
682,018 

750.180 
471.314 
835.634 
771,132 
529.937 
409.073 


Chicago 


Philadelphia 

St. Louis 


Boston r 


Baltimore 


Cleveland 

Buffalo 


San Francisco 

Cincinnati 

Pittsburg 


New Orleans 

Detroit 


Milwaukee 

Newark 


Washington 

Jersey City ....... . 

Louisville 


Minneapolis 

Providence 

Indianapolis 

Kansas City, Mo. . . 
St. Paul 


Rochester 


Denver 


Toledo 


Alleerhenv 


Columbus 

Worcester 


Syracuse ... 




Total 















Problems like the following may be made from the table, and 
assigned to different pupils. 



42 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

1. Find the totals of all columns of the table for the cities 
whose inhabitants numbered over 500,000 at the 1900 census. 

2. Find the totals for the cities whose populations in 1900 were 
between 250,000 and 500,000; for cities with populations between 
100,000 and 250,000. 

Note. — These intervals may be shortened or lengthened at will. Indi- 
vidual pupils majr do different parts of the work, thus obtaining a large 
number of individual problems. In each case the pupil should tell what 
his total means. 

3. Find the increase in population from 1890-1900 of Cleve- 
land, Ohio; of other cities. 

4. IIow many more teachers in 1900 were there in Minneapolis 
than in Louisville? in Xew York City than in Chicago? in Boston 
than in St. Louis? in Chicago than in Boston? 

5. IIow many more pupils in 1900 were enrolled in the schools 
of Boston than in those of St. Louis? than in those of Balti- 
more? of Cleveland? 

6. IIow much more money was expended in 1900 on schools 
in Boston than in St. Louis? than in Cleveland? IIow much more 
in Cleveland than in Baltimore? than in Buffalo? IIow much 
more in Pittsburg than in Cincinnati? 

7. IIow do the combined school expenditures for Ifew York 
City, Chicago, Philadelphia, and Boston compare with the com- 
bined school expenditures of all the rest of the cities in the table? 
IIow do those of Xew York City and Chicago compare with the 
total of all the rest? 

8. Find the totals for the five columns of the entire table. 
What is the increase in the total population of these cities from 
1890 to 1900? 

§30* Subtraction of Literal Numbers. 

1. How many five-cent pieces are 7 five-cent pieces — 3 five- 
cent pieces? 

2. How many dimes are 12 dimes — 5 dimes? 

3. How many 9's are 16 9's-9 9's? 

4. How manv c's are I3c - 7c? 



MULTIPLICATION 43 

5. Write the differences : 

18a; 26y 43^ 682? 683a 1021«? 
dx Sij d7z 'd9z 97a 879^ 

6. A lot contains 160.T sq. ft., and the house covers 40a; sq. ft.; 
how many square feet of the lot are not covered by the house? 

7. A boy earned 15a; cts. on Friday and spent 10a; cts. on Sat- 
urday ; how many cents did he save? 

8. Tell what number x stands for in these problems: 

(1) 12 -a; =7. (6) 7a; -5a; = 10. (11) 50a; = 100. 

(2) 65 -a; = 20. (7) 16a; - 13a; = 15. (12) ^a; = 50. 

(3) a; - 35 = 60. (8) 9a; - 5a; = 20. (13) fa; = 27. 

(4) a; - 19 = 7. (9) 7a; = 63. (14) fa; = 7. 

(5) 3a; - 2a; = 8. (10) 9a; = 108. (15) Ja; = 63. 
Literal numbers are numbers denoted by letters. 



MULTIPLICATION 
§31. Definitions. oral work 

1. At 75^ a day, how many dollars does a boy earn in 6 days? 

This problem may be solved in two ways. We may say he 
earned the sum of 75^ + 75^ + 75^ + 75^ + 75^ + 75^, which is $4.50. 
Or, we may say he earned 6 times 75^ (6 x 75^), or $4.50. 
In either case his earnings are $4.50. 

2. Which is the shorter way? How do the addends compare 
in the first solution? 

3. Find in two ways how far a vessel will sail in 4 da., making 
22 mi. a day. 

4. A factory employee, working by the hour, makes the follow- 
ing daily record for a week : 8 hr. ; 8 hr. ; 8^ hr. ; 9 hr. ; 9 hr. ; 
9^ hr. How many hours does he work during the week? 

5. How can you solve this problem in more than one way? 
Give reason for your answer. 

6. When the addends are unequal, as In problem 4, what is 
the ouly way they can be combined? 

7. When the addends are equal, as in the first two problems, in 
how many ways can they be combined? Which is the shorter 



44 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

way? By what name do we know this shorter way? What then 
is multiplication? 

MuUij)lication of whole numbers is a short way of finding the 
sum of equal addends when the number of addends and one of 
them are given. 

The given addend is the multiplicand. 

The number of equal addends is the multiplier. 

The result, or sum, is the product. 

The multiplicand and multiplier atq factors of the product. 

The sign of multiplication x is read "multiplied by" when it 
is written after the multiplicand and "times" when it is written 
before the multiplicand. 

Thus 22 mi. X = 132 mi. is read "22 mi. multiplied by 6 
equals 132 mi.," and 6 x 22 mi. = 132 mi. is read "6 times 22 
mi. equals 132 miles." 

An expression like 6 x 22 mi. = 132 mi. is called an equation. 

8. At $3.50 a pair, how much will 12 pairs of shoes cost? 

$3.50 = cost of 1 pair; 
$3.50 X 12 = cost of 12 pairs. 

$3.50 is the multiplicand., 12 is the multiplier .^ and $42.00 is 
the product. 

When a problem is expressed in an equation, the equation is 
called the statement of the problem. 



WRITTEN WORK 

Make statements and solve : 

1. There are 128 cu. ft. in 1 cd. of wood. How many cubic 
feet in 15 cords? 

Statement. 15 x 128 cu. ft. = ? 

Or, 15 X 128 cu. ft. =x cu. ft. 
Find the number which should stand in place of x. 

The number which should stand in place of x in an equation 
is called the value of x. 

2. There are 5280 ft. in a mile. How many feet in 24 miles? 

3. 12 lb. of ham at 17^ per lb. = how many dollars? 

4. A cold wave from the northwest traveling at the rate of 32 
mi. an hour moves how many miles in 24 hours? 



MULTIPLlCATIOil 45 

5. If galvanized telegraph wire weighs 525 lb. per mi. , how 
many pounds of wire will it take to stretch 6 wires from Chicago 
to Milwaukee, a distance of 82 miles? 

6. How much will this wire cost at 8^ per pound? 

7. Find the cost of 275 fence posts at 65^. 

$. 65 = cost of one post ; 
275 = number of posts; 
$.65 X 275 = 1178.75 (cost of 275 posts) is the equation of the problem. 

f^.65 In this problem, which number is the 

_?25 multiplicand? AVhich the multiplier? 
325 
455 
ISO 



$178.75 



This problem may be solved also thus : If each post t2.75 

cost 1^, 275 posts would cost $2.75; but as each post ^ 

costs 65^, the cost of all will be 65 x $2.75, and the {375 

work may be arranged as shown. 2550 

Or, we may say 275 times $.65 is the same as 65 

times $2.75, and multiply as above, $178.75 

Since the numerical product of factors is the same, whichever 
is used as the multiplier, the smaller number is generally taken 
for the multiplier for convenience, 

8. In 64 pk. there are how many quarts? 

9. How many pounds in 494 bu. of corn, if in 1 bu. there are 
56 lb.? how many pounds in 25 bu.? how many in x bushels? 

10. There are 231 cu. in. in 1 gal. How many cubic inches in 
587 gal.? in y gal.? in a gallons? 

§33. Tables. 

These products must be learned thoroughly : 



3X3= » = 


3X 3 


3X4 = 12 = 4X 8 


4 X 3 = 12 = 


3X 4 


4X4=16 = 4X 4 


5 X 3 = 15 = 


3X 5 


5X4 = 20 = 4X 5 


6X3 = 18 = 


3X 6 


6X4 = 24 = 4X 6 


7X3 = 21 = 


3X 7 


7 X 4 = 28 = 4 X 7 


8X3 = 24 = 


3X 8 


8X4 = 32 = 4X 8 


9X3 = 27 = 


3X 9 


9 X 4 = 36 = 4 X 9 


10X3 = 30 = 


3X10 


;0X4 = 40 = 4X10 


11X8 = 33 = 


3X11 


11X4 = 44 = 4X 11 


12 X 3 = 36 = 


3X13 


12X4 = 48 = 4X12 



4C 



RATIONAL ORAVMAR SCHOOL ARITHMETIC 



§34, 



Learn each of these in the same way: 






3X6 = 18 


3X7 = 21 


3X8 = 34 


3X9 = 


27 


4X0 = 24 


4X7 = 28 


4X8 = 32 


4X9 = 


36 


5 X 6 = 30 


5X7 = 35 


5X8 = 40 


5X9 = 


45 


6 X 6 = 36 


6X7 = 42 


6X8 = 48 


6x9 = 


54 


.7X6 = 42 


7 X 7 = 49 


7 X 8 = 56 


7X9 = 


63 


8X6 = 48 


8 X 7 = 56 


8 X 8 = 64 


8X9 = 


73 


9X0 = 54 


9 X 7 = 63 


9 X 8 = 72 


9X9 = 


81 


10X6 = 60 


10 X 7 = 70 


10 X 8 = 80 


10X9 = 


90 


11X6 = 66 


11X7 = 77 


11X8 = 88 


11X9 = 


99 


12 X 6 = 72 


12X7 = 84 

ORAL 


12X8 = 96 
WORK 


12X9 = 


108 



1. 



At current prices, find the cost of the following article^. 
J lb. Oolong tea. 2. 2 erasers. 



3 doz. eggs. 
5 lb. ham. 
3 lb. leaf lard. 

2 lb. Java coffee. 

3 lb. best quality of butter. 



3 tablets linen paper. 

3 packages white enyelopws. 

4 doz. pens. 

2 lead pencils. 

2 bottles writing fluid. 



2 doz. oranges. 
2 doz. lemons. 
1^ doz. bananas. 
1 pk. apples. 



4. 8 lb. rib roast. 

2^ lb. porterhouse steak. 
3 lb. lamb chops. 
2 lb. sausage. 



5. 3 cd. hard wood. 
2 T. hard coal. 
1^ cd. pine slabs. 
4 T. soft coal. 



2 T. hay. 

3 bu. oats. 
2 bu. corn. 

4 bales straw. 



7. 5 lb. tomatoes. 
8 bunches radishes. 
3 pk. turnips. 
1 pk. beets. 
3 qt. Lima beans. 



8. 3 pr. shoes. 

i doz. tennis balls. 
3 baseball bats. 
2 catcher's gloves. 
1 sweater. 



MULTIPLICATION 



47 



§36. Farm Account Keeping. — The forms below show how a cer- 
tain farmer keeps a systematic account of receipts and expendi- 
tures with each of his fields. The fields referred to are those 
shown in the drawing of Fig. 4, p. 7. 



WRITTEN WORK 

The items for which money was paid out or received were 
put down with the dates of payment or of receipt in a day-book 
thus: 

Day-Book for Forty- Acre Cornfield 

1. Paid for 4 da. work removing stalks @ $1.15, Mar. 18. 

2. Paid for 9 da. plowing @ $1.15, Apr. 30. 

3. Paid for 2 da. harrowing @ $1.00, May 1. 

4. Paid for 3 J da. planting corn @ $1.20, May 18. 

5. Paid for 6 bu. seed corn @ 90^, May 18. 

6. Paid for 2 da. replanting corn @ $1.15, May 30. 

7. Paid for 5 da. harrowing corn @ $1.15, May 30. 

8. Paid for 7 da. cultivating ^ $1.15, June 15. 

9. Paid for 6 da. cultivating @ $1.15, June 30. 
10. Paid for 6 da. cultivating @ $1.15. July 15. 

n. Paid for cutting 240 shocks corn @ 8i<^, Sept. 15. 
13. Paid for husking 1392 bu. corn @ 3^, Nov. 20. 

13. Paid for shelling 1336 bu. corn @\^y Jan. 10. 

14. Sold 1336 bu. corn @ 38f2^, Jan. 10. 

15. Received pay for 3 mo. pasture @ $1.50, Feb. 28. 

1. These items are here arranged in the form of a receipt and 
expenditure account. Fill out the vacant columns, and find the 
totals and the net profit for this forty-acre cornfield: 



In Account tvith Forty-Acre Cornfield 
Expenditures Receipts 



Mar. 18 

Apr. 30 

May t 

" 18 

•' 18 
•• 30 



June 15 

'• 30 

July 15 

Sept. 15 

Nov. 20 
Jan. 10 



Removing stalks, 4 da., @ 

$1.15 
Plowing, 9 da., ® $1.15 
Harrowing, 2 da., @ $1.00 
Planting com, 3% da., @ 

Seed corn, 6 bu., @ 90^ 
Replanting, Si da., (g^ $1.15 
Hawowlng corn, 5 da., @ 

$1.15 

Cultivating, 7 da., ® $1.15 

6 da., @, $1.15 

6 da., @ $1.15 

Cutting 240 shocks corn (^ 

8^0 
Husking 1393 bu. corn @. 30 
Shelling 1336 bu. com (^ 

Total 



Jan. 10 
Feb. 28 



1336 bu. com @ 380 
3 mo. pasture @ $1.50 

Total 

Expenditures to be de- 
ducted 
Net profit from 40 acres 
'* I)eracre 



Account closed March 
1901 



48 



RATIOKAL GRAMMAR SCHOOL ARITHMETIC 



2. From the following list of day-book items for the twenty- 
acre wheatfield the account below is arranged. Fill out the vacant 
columns and find the totals, the net profit on the whole field, and 
the net profit per acre. 

Day-Book for Twenty-Acre Wheatfield 

1. Paid for 4 da. plowing @ $2.50, Oct. 20. 

2. Paid for 3 da. harrowing and rolling @ $2.25, Oct. 25. 

3. Paid for 25 bu. seed wheat @ $1.25, Oct. 25. 

4. Paid for 21 da. drilling wheat @ 12.50, Oct. 30. 

5. Paid for cutting 20 acres wheat @ 75^, July 15. 

6. Paid for 2 da. shocking wheat @ $1.75, July 15. 

7. Paid for threshing 480 bu. wheat @ 6<', Aug. 80. 

8. Paid for 3^ da. help in threshing @ $1.50, Aug. 30. 

9. Sold 425 bu. wheat @ 68<^, Dec. 22. 

10. Received pay for 4 mo. pasture @ $1.50, Mar. 2. 

In Account with Twenty-Acre Wheatfield 

EXPENDITURBS RBCBIPTS 



Oct. 20 
" 25 



July 15 
•• 15 

Aug. 30 
•• 30 



Plowing, 4 da., @ $2.5') 
Harrowing and rolling, 

3 da., @$2.25 ' 
Seed wheat, 25 bu., % $1.25 
Drilling, 2% da., ® $2.50 
Cutting 20 acres wheat @ 

75^ 
Shocking wheat, 2 da., @ 

$1.75 
Threshing 480 bu. wheat 

Help threshing, 3^ da., @ 
$1.50 

Total 



Dec. 22 
Mar. 2 



Wheat, 425 bu., ^ %^t 
Pasture, 4 mo., @ $1.50 

Expenditures to be de- 
ducted 
Net profit from the field 
" '* i)er acre 



3. Draw up these facts into an account, and find the totals and 
the net profit, and treat in the same way as above : 



Day- Book for Thirty- Acre Pasture 

1. Bought 120 bu. corn @ 37^, Apr. 23. 

2. Sold 5 yearling steers @ $18.00, Apr. 25. 

3. Sold 3 yearling heifers @ $15.00, July 6. 

4. Sold 2 milk-cows @ $46, July 8. 

5. Bought 4 two-yr. old heifers @ $25, July 10. 

6. Sold 8 hogs weighing 2064 lb., @ $5 per 100 lb., July 20. 

7. 8 head of hogs, worth $12 each, died July 25. 

8. Bought 20 pigs @ $2.50, Aug. 10. 

9. Sold 3 two-yr. old colts @ $72, Sept. 30. 

10. Sold 2 draught horses @ $195, Sept. 30. 

11. Sold 9 four-yr. old steers @ $68, Oct. 25. 

12. Bought 10 year ling calves© $12, Nov. 1. 

13. Sold 3 Jersey milk*o>ows @ $45, Nov. 15. 

14. Paid 6 mo. wages for one man @ $80, Dec. 28. 

15. Sold 8 head of hogs @ $15, Jan. 10. 

16. Bought 12 head of hogs @ $9. 50, Feb. 14. 



MULTIPLICATIOK 



49 



4. Fill out this account of the fifty-acre oatfield. 
Expenditures » Receipts 



Mar. 30 

" 20 
" 20 

Aug. 18 

*• 18 

Sept. 15 
•» 15 



RemoYing old stalks 5 da. 

(^$1.86 
100 bu. seed oats @ 35^ 
61 da. work sowing oats 

@ $1.25 
Harvesting 50 acres oats 

4| da. labor in harvesting 

@ $1.75 
Threshing 2148 bu. (gj S0 
5 da. help threshing @. 

$1.50 

TOTAIi 



Feb. 15 
" 20 



1800 bu. oats @ 2210 
85 loads straw ^; $2.75 

Expenditures to be de- 
ducted 
Net profit from 50 acres 
" '* per acre 



5. Treat the meadow account similarly. 

In Account with Ten- Acre Meadow 



Expenditures 



Receipts 



Nov. 20 

Aug. 30 

" 80 

" 30 



10 bu. timothy seed (^ 

$2.00 
Cutting 10 acres hay @ 

35e 
Raking and shocking hay 

5da. @ $1.50 
Stacking 15 tons hay @ 55^ 



Dec. 15 



12 tons hay ^ $9.50 

Expenses to be deducted 
Net profit from 10 acres 
** " per acre 



6 . Draw up the following items into an account like the one 
above; and find totals and profit or loss to the farmer. 
Day-Book for House and Bam Lot {10 acres). 



1. Apr. 15. Bought 6 bu. seed po- 

tatoes @ $2 

2. " 20. Bought 30 pkg. gar- 

den seeds @ 10^ 
a " 20. Bought 4 pkg. garden 
seeds @ 25^ 

4. May 15. Paid 5 da. wages @ 

$1.25 

5. " 25. Sold 100 bunches on- 

ions @ 21 f 

6. " 30. Sold 148 bunches cel- 

ery @ 3^ 

7. •• 30. Sold 200 bunches let- 

tuce @ 2i^ 

8. " 30. Sold 240 bunches as- 

paragus @ Sf 

9. June 30. Sold 14 bu. peas @ ¥ 

per qt 

10. '* 30. Sold 28 bu. green beans 

@ 4^ per qt. 

11. *' 30. Sold 300 bunches as- 

paragus @ 5^ 

12. •* 80. Paid 18 da. wages @ 

11.25 

13. July 31. Sold 10 bu. new pota- 

toes @ 11 



14. July 31. Paid 15 da. wan;es @ 
$1.50 
" 31. Sold 50 heads cabbage 

@ 8<25 
'' 31. Sold20doz. ears sweet 
corn @ 8,^ per dozen. 
Aug. 31. Sold 140 lb. tomatoes 



15. 
16. 
17. 
18. 
19. 
20. 



31. Sold 10 bu. early apples 

@75^ 
31. Sold 25 heads cabbage 

@7^ 
31. Sold 15 bu. peaches @ 

$1. 

21. ** 31. 10 doz. ears corn @ If 

22. ** 31. Paid 20 da. wages @ 

$1.50 

23. Sept. 30. Sold 55 bu. potatoes @ 

65 

24. •' 30. Sold 20 bu. Lima beans 

@ $1.25 

25. " 30. Paid 20 da. wages© 

$1.25 

26. Dec. 20. Sold 30 bu. apples @$1 

27. " 30. Sold 10 bbl. (40 gal. 

each) cider vinega'' 
@ 30/ per gal. 



60 RATIONAL ORAMMAR SCHOOL ARITHMETIC 

7. Make a statement from these items of general expense, 
not included in any of above accounts, and find the total profit or 
loss : 

1. Apr. 20. Bought 3 sets of har- 7. Aug. 30. Built corn crib, cost 

ness @ $28 , 1112.00 

2. " 25. Paid tax on 160 acres® 8. Sept. 10. Paid for 120 mo. pas- 

25^ turing stock @ $1. 50 

3. *• 30. Bought 1 wagon @ $60 9. Feb. 28. Paid for 45 mo. pas- 

4. • ' 30. Bought 2 plows @ $35 turing stock @ ^1. 25 

5. May 10. Bought 3 cultivators @ 10. " 28. Paid 30 da. fertilizing 

$25 @ $1.25 

6. Aug. 20. Paid 84 rods tile ditch- 11. " 28. Paid 8 da. mending 

ing @ 50,<^ fence @ $1.25 

8. Add all the net receipts and all the net expenditures for 
the separate fields as given in the records of these fields in prob- 
lems 1 to 7 above. Find the net earnings for the year of 
the whole farm. 

§36. Problems. 

1. A double eagle weighs 516 grains, and a gold dollar 25.8 
grains. Find the difference in weight between a double eagle and 
2 gold dollars. 

2. AVhat will a peck of peanuts cost at 5^ a pint? 

3. At $50 a front ft., what will 50 ft. of city land cost? 

4. If milk costs 0^ a qt. and I buy 2 qt. a day, what is my milk 
bill for April? 

5. The average milk yield of a cow was 6 qt. a da. for 120 da. 
If this milk was all sold @ G^-^ a qt., how much did the owner 
receive for it? 

G. The cow's feed cost the owner $2.50 per mo. of 30 da. 
How much profit did the owner receive from the cow's milk during 
the 120 days? 

7. A cow gives 12 lb. of milk a da. and the butter made from 
this milk equals ^V ^^ the weight of milk. How many pounds of 
butter does the milk furnish in 30 days? 

8. If butter is selling for 35^ a lb., what is the butter yield of 
this cow worth in 30 days? 

0. G8 qt. of a certain Jersey cow's milk yield 7 lb. of butter. 
If milk is selling @ 6^ and butter @ 35^, is it more profitable to 



MULTIPLICATION 51 

sell 68 qt. of the milk of this cow, or to make butter of it and sell 
the butter? How much more profitable is it? 

10. The pulse of a healthy man beats 72 times per minute. 
The heart contracts once for each pulse beat. How many times 
do the muscles of a healthy man's heart contract in 1 hr.? in 
1 da. at the same rate? 

11. Count your own pulse beats for 1 min. and find how manj 
times your heart beats in a day at this rate. 

12. Tie a heavy ball to a fine string, or split a bullet and 
fasten it to a thread, and suspend the string to a hook or nail 
so that 1^ ft. of it may vibrate freely. Start it swinging and 
count the number of vibrations per minute. At the same rate 
how many timeS would the ball vibrate in 24 hours? 

13. Solve the same problem with the suspending string 3 ft. 
long; with the string 3 ft. 3 in. long. 

14. My watch ticks 270 times per minute; how many times 
does it tick in 1 hr.? in 1 day? 

15. Light travels 186,600 mi. per second ; how far does it travel 
in 1 day? 

16. A man smokes 3 cigars a day, and pays 15^ apiece for 
them. How much does this add to his expense account in 365 
days? 

§37. Multiplying by Factors. 

1. 7x8 = 56 28x2 = 56 14x4 = 56 
7, 8, 28, 2, 14, and 4 are all factors of 56. 

2. Write the factors of 44, 48, 96, 35, 84, 81, 49, 108. 

3. The two equal factors of 144 are 12 and 12. Give the two 
equal factors of 4, 9, 25, 49, 36, 121. 

4. 8x2 = ? 8x2x5 = ? 8x10 = ? 

5. 12x2 = ? 12x2x3 = ? 12x6 = ? 

6. 20x4 = ? 20x4x3 = ? 20x12 = ? 

7. 25x4 = ? 25x4x6 = ? 20x24 = ? 

8. 64x9 = ? 64x9x7 = ? 64x63 = ? 

9. Instead of multiplying a number by 15, by what two num- 
bers in succession may I multiply it and get the same product? 



52 SATIOXAL 6HAHMAR SCHOOL ARITHMETIC 

10. Multiplying any number by 21 gives the same product as 
multiplying it by what two numbers in succession? 

11. 50 X 2 X 4 gives the same product as 56 multiplied by what 
single number? 

12. Multiplying a number by several factors one after another 
gives the same product as multiplying it by what single number? 

A number which has factors other than itself and 1 is called 
a composite number. A number such as 3, 5, 17, etc., which has 
no factors other than 1 and itself, is 9, prime number. 

13. Instead of multiplying a number by a composite number, 
in what other way may I multiply it to obtain the same product? 

14. Which of these numbers are prime, and which composite : 
24, 25, 19, 6, 2, 21, 23, 84, 45, 43, 27, 28? 

§38. Multiplying by 10, 100, 1000, 10,000. 

1. 84x10 = ? 84x100 = ? 84x1000 = ? 84x10,000 = ? 

2. 327x10 = ? 327x100 = ? 327x1000 = ? 327x10,000 = ? 

3. How may any whole number be quickly multiplied by 10, 
100, 1000, or 10,000? 

§39. Multiplying by a Number Near 10, 100, 1000, 10,000. 

1. 746x9 = ? 

Solution.— This is once 746 less than 10 X 746, or 7460 - 746 = 6714. 

2. 9G5 X 99 = ? Ans, 96,500 - 965 = 95,535. 

3. 965 X 98 = ? Ans. 96,500 - 1930 = 94,570. 

4. 965 X 101 = ? Ans. 96,500 + 965 = 97,465. 

5. 965 X 102 = ? 

6. 873 X 11 = ? 

7. 8473 X 1001 = ? 

8. 8473 X 999 = ? 

§40. Multiplying by 25, 50, 12i, 75, 500, or 250. 

1. 65x 25 =? 25 =iof 100. i of 65x100 = 1625. 

2. 83x 50 =? 50 =^of 100. ^ of 83x100 = 4150. 

3. 638 X 12^ = ? 12i = i of 100. i of 638 x 100 = 7975. 

4. 132 X 75 =? 75 = J of 100. J of 132 x 100 = 9900. 

5. 640 X 500 = ? 500 = i of 1000. 

6. 740x250 =? 250 =i of 1000. 



MULTIPLICATIOK 53 

We see from problem 1 that to multiply by 25, we may multi- 
ply by 100 and take i of the product. 

7. Make a rule, different from the ordinary rule, for multiplying 
quickly by 50; by 12^; by 250; by 500. 

8. Make a rule for multiplying quickly by 33^; by 333 J. 
This shows the importance of remembering that : 

i of 100 = 50 J of 1000 = 500 

J of 100 = 25 i of 1000 = 250 

J of 100 = 75 f of 1000 = 750 

i of 100 = 121 i of 1000 = 125 

|oflOO = 37J i of 1000 = 375 

i of 100 = 621 I of 1000 = 625 

i of 100 = BH i of 1000 = 875 

,3^ of 100= 6i Aofl000 = 62i 

i of 100 = 33J i of 1000 = 333J 

i of 100 = 661 i of 1000 = 6661 

9. Make rules for multiplying quickly by the numbers on the 
right hand side of these equatioQs. 

§41. Multiplying When Some Digits of the Multiplier Are Factors 
of Others. 

1. From twice a number, how may you obtain 4 times the same? 
From 4 times a number, how obtain 8 times the same? 

2. Multiply 19,279 by 842. 

SonjnoN.— 19279 

842 

38558 = 19279 X 2, the first partial product 
771160 = 20 times the first partial product 
15423200 = 20 times the second partial product 

16232918 

3. Observe the relations of the digits of each multiplier below, 
and shorten the work of multiplication, as above. Prove work : 

4,783 X 363 29,847 x 248 

12,875x442 86,729x393 

§42. Checking Multiplication. 

Any of the above ways may be used to test the correctness of 
products obtained in the usual way. 



l_j_ ; 

V 



5t RATIONAL GRAMMAR SCHOOL ARITHMETIC 

To check by casting out the nines, cast the nines out of the 
multiplicand, the multiplier, and the product. Multiply the 
excesses of the multiplicand and the multiplier. Cast the nines out 
of this product. If this last excess e,quals the excess of the 
original product, the work is probably correct. 

Illustration.— -To check 6848x619 = 4238912, excess in 6848 = 8; 
excess in 619 = 7; excess in 4238912 = 2; 7x8 = 56, excess in 56 = 2. 
As excess in 4238912 and in 56 are equal, the work is checked. 

A very useful check against blunders is to examine the facts of 
the problem and to decide about what the answer must be, before 
beginning to solve it. 

§43. Multiplying by Fractional Nambers. 

1. How inauy square yards in a room 

4 yd. X Si yards? (Here x is read "by,") 

Solution. — How many sq. yd. fill a strip 
'g 1/ 1^ ' — I — LJ 1 yd. wide extending along the long side? 
^**^ ** How many such strips cover the floor? 8| sq. 

FiGUBK 15 yd. X 4 = a: sq. yd. What is a:? 

2. What is the cost of 68^ acres of land @ $80? 

Solution.~$80x68=$5440, and J of 180 = $40. Then 80X68i = 
15480. Observe that $80 X 68} means 80 X 68+ i of 80. 

3. Tell what these problems mean and then solve them: 

$G4xl6i = ? 811b. x25i = ? 24 ft. x8f=? 

5280 mi. x 14| = ? 807 yd. x 19f = ? 5^^ sq. yd. x 6 = ? 

160 A. X 12| = ? 5i sq. yd. x 5i - ? 

Note. — Solve the last problem by drawing a square 5i units long and 
5) units wide, and dividing it up into square units. 

The problems just solved show that J x 8 means J of 8, or 3 
of the 4 equal parts of 8. 

4. Give the meaning of these problems and find the value of 
the letter in each : 

ixd=x, fx21 = .c. ix66 = y. iixl08=y. ^x76 = z. 

§44. Suggestions for Problems. 

Make and solve problems based on the following facts : 
1. A steel rail weighs G4 lb. per yd. of length. The distance 
from Washington to Baltimore is 42 mi. ; to Philadelphia, 138 
mi. ; to New York, 227 mi. ; to Boston, 459 miles. 
Note— There are 1760 yd. in 1 mile. 



MULTlPLTrATIOK 00 

2. There are double tracks between Washington and each of 
the cities mentioned. 

3. The distance from New York to San Francisco is 32G2 
lUiles. 

4. The distances in miles from Chicago to 14 railroad centers 
of the United States are given here: 

Indianapolis '.184 Rochester ft05 

St. Louis 383 Baltimore 801 

Cincinnati 298 Washington 820 

Cairo 364 New Orleans 912 

St. Paul 410 New York 913 

Omaha 490 Denver 1028 



Buffalo 536 San Francisco 

5. Galvanized telegraph wire weighing 572 lb. per mi. is nsed 
for distances over 400 mi., and for distances under 400 mi., wire 
weighing 378 lb. per mi. is used. 

6. 8 wires run between Chicago and Xew York. 

7. Galvanized iron telegraph wire costs 0^ per pound. 

8. Sound travels in water at the rate of 4708 ft. per second ; 
in air 1130 ft. per second. 

9. Silver is worth 56^ an oz., 12 oz. to the pound. 

10. Hay costs $23 per T. ; oats 42^ per b«. A horse is 
fed 162 lb. of hay and 10 bu. of oats a month. Bedding costs 
$2 a month. 

11. Standard silver is ^V P^^^ silver and ^V copper. A silver 
dollar weighs 412.5 grains. The total number of silver dollars 
coined on this basis was 378,160,769. 

12. The number of grains in an ounce of silver is 480. The 
amount of silver bought under the Sherman Law by the United 
States government to coin into money was 168,674,682 ounces. 

13. Light travels 186,600 mi. per second. It requires 448 sec- 
onds for light to reach the earth from the sun. 

14. It reaches the earth from the moon in 1^^ seconds. 

15. An 02. of pure gold is worth $20.67. There are 12 oz. in 
1 lb. of gold. 

16. An Alaskan miner can take away 200 lb. from the mining 
district. 



56 



BATIONAL GRAMMAB SCHOOL ARITHMETIC 



17. A cu. ft. of granite weighs 170 lb. A granite step meas- 
ures i' X 2' X 8'. 

18. The distance around the driving wheel of a locomotive 
engine is 22 ft* In going a certain distance the driver turned 
5280 times. 

19. In the year 1901, 82,305,924 lb. of tea and ^511,041,459 
lb. of coffee wore imported into the United States. Tea was 
worth 48^ and coffee 2G^ per pound. 

20. A prize- winning steer weighed 16.03 cwt. (1 cwt.= 100 lb.) 
and sold for $9.00 per hundredweight. 

21. Another steer of the same lot weighed 1622 lb. and sold 
for $8.85 per hundredweight. 

22. A boy bicyclist rode a miles per da. for h days. 

23. Pupils should prepare and solve problems based upon price 
lists obtained from the grocer and the butcher (see p. 4), or from 
the market reports of the daily papers. 

§46. Eainfall. — How long is a cubic inch (Fig. 16)? how wide? 

how high? What is a cubic foot? 

A. cubic yard? ^X^\^ <^^*^^ 





Figure 16 



Figure 17 



A tin box 3 in. square on the bottom and 9 in. high (Fig. 1?) 
was used by a school as a rain gauge. A second tin box 1 in. 
square on the bottom and 9 in. high was used to measure the 
depth of water in the large box. After the water was poured 



MULTIPLICATION 



57 



2 foot 



from the large box into the small box, the depth of water in the 
small box was measured with a thin stick and a foot rule. 

1. The rain gauge was placed one evening where the rain could 
fall freely into its open top. During the night it rained and the 
next morning the water was poured from the gauge into the small 
box. It filled the small box to a depth of 9 in. How deep did 
the water fill the large box? How many cubic inches of water 
were caught in the large box? 

One cu. in. of water for every sq. in. of surface is what is 
mQant by 1 in. of rainfall. What is 6 in. of rainfall? 

2. How many cu. in. of water are there in a layer 1 in. deep 
in the large box (Fig. 17)? 2 in. deep? 5 in. deep? 9 in. deep? 

3. How many cu. in. of water fell on 1 sq. 
ft. of the ground during a rainfall of 1 in. 
(Fig. 18)?' of 2 in.? of 3 in.? of 6 inches? 

The number of cubic units (cu. in., cu. ft., 
cu. yd., etc.), a vessel holds, when full, is called 
its capacity. • 

4. What is the capacity of a square- 
cornered box 3" X 3" X 9"? 

5. What was the depth of rainfall during a 
shower if the large box caught enough water to fill the small box 
half full? to a depth of 3 in.? of 6 in.? of 1 in.? of 2 in.? of 7 
inches? 

6. How many cu. in. of water fell on 1 sq. ft. of the ground 
during a shower giving 1 in. of rainfall? 2 in.? ^ inch? 

7. During a rainfall of 1 in. how many sq. ft. of ground 
received enough water to make 1 cu. ft.? 2 cu. ft.? 12 cubic 
feet? 

8. During June, 1902, the rainfall in the vicinity of Chicago 
was 6i in. During this month how many cu. in. of water fell 
on 1 sq. in. of ground? on 1 sq. ft.? on 12 sq. ft.? on 1 sq. 
yd.? on 30 sq. yd.? on ^ sq. yd.? on 30^ square yards? 

9. Just before a shower set an uncovered bucket where the 
rain may fall freely into it. After the shower measure the 
depth of the water in the bucket to find the depth of rainfall. 



s" a" s*' s" 



Figure 18 



58 RATIONAL (JRAMMAR SCHOOL ARITHMETIC 

Find how many en. in. or cu. ft. of water fell on each sq. ft.? 
each sq. yd.? each sq. rd. of the ground? 

Note. — The bucket used must have the same size at the top and bot- 
tom, with straight sideS. Why? 

10. During the first 9 mo. of the year 1902, the region about 
Chicago received 32 in. of rainfall. IIow many cu. ft. of water 
fell during this time on a garden bed 10' x 14'? on a garden 48' x 
84'? over a city block 250' x 350'? 

11. If the walls of your schoolroom were water-tight, how 
many cu. ft. of water would it hold if it were filled 1 ft. deepT 
2 ft. deep? 5 ft. deep? 8 ft. deep? to the ceiling? 

12. Find the areas of these rectangles : 

r/'x8", 12"x28", 40'x64', 
376' X 486', 9' x x\ 9 yd. x a;yd., 
a mi. X }> mi., x units x y units. 

How can you find the number of square units in any rectangle? 

Note.— a? X y is written ocy and read *'a:, y.** « 

13. How many cubic feet are there in 1 cu. yd.? in 18 cu. yd.? 

14. What is the capacity of a square-cornered box 3" x 3" 
x 9"? 3' X 3' X 9'? of a square-cornered room or space, 3 yd.x 3 yd. 
X 9 yd.? 3 rd. X 3 rd. x 9 rd.? 3x3x9? 3 x 3 x a? 

15. Find the capacity of a square-cornered box 3" x 4" x V \ 
3"x4"x8"; 3'x4'x8'; 3'x4'xl5'; 3 yd. x 3 yd. x 3 yd.; 3 
units X 4 units x 15 units; 3x4xir; 3xa;xy; a units x J units 
X c units. 

Note. — ^The product a? X 2^ X « is written xyz and read "a?, y, «." 

16. How can you find the number of cubic units in any square- 
cornered vessel? 

17. Find the total weight of a snow load of 25 lb. per sq. ft. 
on a flat rectangular roof 25' x 48'. Find the weight of a load 
of 15 lb. per square foot. 

18. Find the total weight of the shingles and sheathings for 
both sides of the roof shown in Fig. 19, if shingles weigh 3 lb. 
per sq. ft. , and sheathing 5 lb. per square foot. 

Note. — The eaves project 2 ft. over the plate. 



MULTIPLICATION 



59 



19. Find the weight of the snow load on both Bides of 
the roof, the weight on each sq. ft. of surface being 12 pounds. 

20. Find the cost at 3^ per sq. yd. of lathing and plastering 
the four walls and the ceiling, 
no allowances being made. 

21. What is the area of the 
square ABCD of Fig. 19? What 
is the area of the triangle DE(Tt 

22. A strong gale, giving a 
pressure of 16 lb. per sq. ft., 
blows squarely against the end of 
the building. What is the total 
wind pressure against the end including the gable? 

23. How many cu. ft. of space are inclosed by the walls to 
the base, Z>C7, of the gables? 




FIGUBB 19 



§46. Algebraic Problems. 

4 X ic is written ix. If ix = 24, what is the value of a;? 

Notice that the equation 4a; = 24 is the statement of this prob- 
lem. A boy reads 24 pages of a book in 4 days ; how many pages 
does he read a day? What does x stand for in this problem? 

1. What number does x stand for in these equations? 

(l)3a;=18. (4) a;- 8 = 15. (7)a;+ 9 = 17. (10) 5a; + 2a: = 70. 

(2) 6a; = 48. (e5)a;-18= 4. (8) a; 4- 6 = 22. (11) 9a;- 3a; = 36. 

(3) 8a; = 72.' (6) a? - 9 = 8. (9) a; + 16 = 25. (12) 8a; + 5a; = 26. 

2. Write the sum of a and J. 

3. Write the difference of a and h, 

4. Write the product of a and })\ oi a and I and c; of a; and 
y and z, 

5. Answer the following questions if a = 9, d = 8 and c = 6 : 
4« = ? 12S = ? 9c=? «d = ? 4aJ = ? alc^'t Zdbc^t 

6. In the shortest way you can, write eight times a;; seven times 
y; twenty -five times a times 5; a times x times y. 



f)0 EATIONAL GRAMMAE SCHOOL ARITHMETIC 

7. In the shortest way, write and read a times b\ c times x; 
fifteen times a times b times x. 



DIVISION 



§47. Diyitfion and Subtraction Compared. — oral work 

1. A man owes a debt of $12 which he is to pay by work 
at $2 per day. How much will he owe at the end of the first day? 
of the second day? of the third day? of the fourth day? the fifth? 
the sixth? 

2. How long will it take to cancel the debt? 

3. How many times may 2 be subtracted from 12, leaving no 
remainder? How many 2's are there in 12? 

4. A man buys a horse for $90 and is to pay $15 a month 
until the horse is paid for. How much does the man owe after 
the first payment? after the second? the third? fourth? fifth? 
sixth? 

5. How many months will it take to pay for the horse? 

6. How many times may 15 be subtracted from 90, leaving no 
remainder? How many 15's in 90? 

7. What is one of the 6 equal parts of 12? of 90? 

8. A rectangular plot of ground 8 yd. wide by 18 yd. long is 
covered with bluegrass sod. How many square yards of sod does 
it contain? After a strip of sod 1 yd. wide, extending the length 
of the plot, has been removed, how many square yards of sod 
remain? 

9. How many square yards remain after the removal of 2 such 
strips? of 3? of 4? of 5? of 6? of 7? of 8? 

10. How many times may 18 sq. yd. be subtracted from 144 
sq. yd., leaving no remainder? How many 18's are there in 144? 
What is one of the 8 equal parts of 144? 

written work 
1. There are 160 bricks in a pile; find by subtraction how 
many loads of 20 bricks each there are in the pile. How mauv 
20's are there in 160? 



DIVISIOK 61 

2. Find by successive subtraction how many SB's there are in 
616. What is one of the 7 equal parts of 616? 

3. Find by subtraction how many months of 30 da. there are in 
270 da. What is one of the 9 equal parts of 270? 

. 4. What number may be subtracted 4 times in succession from 
120, leaving no remainder? 

5. Find by subtraction how many times $1263 is contained in 
$5052. 

6. Tell bow to find by subtraction how many times one num- 
ber is contained in another. 

7. Find in a shorter way how many 12's there are in 156. 

8. Find in a shorter way than by subtraction one of the 14 
equal parts of 224. By what name do you know this short way? 

9. In the same way find one of the 15 equal parts of 225 ; of 
210; of 240. 

10. A lawn-mower cuts a strip 18 in. wide. How many 
strips must be cut to mow a lawn 18 ft. wide? 20 ft. wide? 30 
ft. wide? 

Division is a short way of subtracting one number from 
another a certain number of times in succession. 

§48. Diyision and Multiplication Compared. — oral work 

1. A horse traveled 72 mi. in 8 hr. ; find the number of miles 
traveled per hour. 

2. 4 bu. potatoes cost $2.40. What was the price per bushel? 

3. A room 4 yd. wide contains 24 sq. yd. ; what is the length 
of one side? 

4. 15 lb. sugar cost 90^; what is the price per pound? 

5. At 1^6 per ton, how many tons of coal can be bought for 
$180? 

6. A floor containing 132 sq. ft. is 11 ft. wide; what is the 
length? 

7. Find the cost of 10 pk. apples @ 14^ per peck. If 10 pk. 
apples cost $1.40, what was the price per peck? 

8. How many pecks of apples @ 14^ can be bought for $1.40? 

9. A train runs 420 mi. in 12 hr. ; find the average number 
of miles per hour. 



62 BATIOKAL GRAMMAR SCHOOL ARITHMETIC 

10. I paid 96^ for veal at 12^ per pound; how many pounds 
did I buy? 

11. What is the cost of 3 T. hay at $12 per ton? 

12. At $12 per ton, how many tons of hay can be bought 
for $36? 

13. A man paid $36 for 3 T. hay; what was the price per ton? 

14. Find the cost of 25 bbl. (barrels) flour at $5. 

15. Paid $125 for 25 bbl. flour; what was the price per barrel? 

16. Paid $125 for flour at $5; how many barrels were bought? 

17. The two factors of a number are 13 and 7; what is the 
number? 

18. The product of two numbers is 91 and one of the numbers 
is 13; what is the other number? 

19. The product of two numbers is 240 and one of the num- 
bers is 20 ; what is the other? 

20. 20 X a: = 240 ; what number does x stand for? 

21. Tell what number the letter stands for in each of these 
equations: 

12^ = 96; 8a; = 56; 7a = 56; 9J = 72; 10« = 150; 24m = 240. 
Division is a way of finding one of two numbers when their 
product and the other number are given. 
The product is called the diiridend. 
The given number is the divisor. 
The required number is the quotie7it. 

Illustration.— The product of two factors is 490, and one of the fac- 
tors is 7. What is the other factor? 

70 
7)490 

"We may also say that the dividend is the number to be 
divided, the divisor is the number by which the 4ividend is 
measured or divided, and the quotient is the measure. 

The sign h- of division is read "divided by," as 60 -*• 12 = 5. 

60 -i- 12= 5 may also be written in the following ways: 

5. 12)60(5 00^^ , 

12)60 ^ 12 ^ 

'The first and second are in common use in division. 
In the third, the line placed between two numbers shows that 
the number above it is to be divided by the number below it; as 



DIVISION 63 

in ^, 1 is the dividend and 3 the divisor. The fraction itself is the 
quotient. The line between the two numbers is the division sign. 
The fourth sign is called the solidus {sol'i-dus). 

§49. Short Division. 

1. How many gallons are there in 296 pints? 

Solution. — As there are 8 pt. in 1 gal. there are as many 37 

gallons in 296 pt. as there are 8's in 296. g )296 

2. How many days in 120 hours? 

3. At $8 per ton, how many tons of coal can be bought for 
$240? 

4. 984 marbles are distributed equally among a certain number 
of boys. Each boy has 82 marbles. There are how many boys? 

5. Selling at 6 for a cent, how much will a dealer receive for 
540 marbles? 

6. A man paid $2.60 for 4 bbl. of lime. What did each bbl. 
cost? 

7. I bought 6 lb. of butter for $1.62. What was the price per 
pound? 

8. A dressmaker used 84 yd. of cloth for 6 dresses, allowing 
the same amount for each dress. How many yards in each? 

9. A train ran 150 mi. in 6 hr. Not allowing for stops, what 
was the average number of miles per hour? 

Find the value of z in problems 10 and 11: 

10. 48 qt. = X gallons. 

11. At 6^ I can buy x lb. of sugar for $1.50. 

12. 9 papers of needles cost 72^. One paper costs how many 
cents? 

13. In 1901, the number of trains entering Chicago every 24 
hr. was about 1320; what was the average number per hour? 

ORAL WORK 

What does x stand for in each of the following equations? 

1. 63-^7 = .'C 4. 45^3 = a; 7. 120-^3 = x 

2. 630 -s- 9 = re 5. 96 -^ 8 = 2; 8. 108 + 9 = a; 

3. 48-*- 4 = a; 6. 960 + 8 = a; 9. 72 -- 6 = a; 



64 RATIONAL 0RA3IMAR SCHOOL ARITHMETIC 

When dividend and divisor are small numbers, the quotient is 
readily seen. We say it is obtained by inspection. 

When the divisor is a large number, the quotient is not readily 
found by inspection. 



§50. Applications. written work 

1. In 61,448 qt. how many pecks are there? 

SoLXTTiON. — In 51,448 qt. there are as many pecks as there are 8 qt. in 
51,448 quarts. 

8 is contained in 51,000, 6000 times, with a remainder of 3000, Write 
6 in the thousands place in the quotient. 3 thousands = 30 hundreds; 30 
hundreds and 4 hundreds = 3400. 8 is contained in 3400, 400 
6431 times, with a remainder of 200. Write 4 in the hundreds 
8)51448 place in the quotient 200 = 20 tens ; 20 tens and 4 tens = 24 
tens. 8 is contained in 24 tens 3 tens times, without a remain- 
der. Write 3 in the tens place in the quotient. 8 is contained in 8 units 
1 unit time. Write 1 in the units place in the quotient. 
In 51,448 qt. there are 6431 pecks. 
Check: 6431 X 8 = 51,448. 

2. There are 5280 ft. in a mile ; how many yards are there in 
1 mi.? in 2 miles? 

3. If limestone weighs 160 lb. per cubic foot, how many cubic 
feet are there in a piece of limestone weighing 6400 pounds? 

4. If marble weighs 170 lb. per cubic foot, find the number of 
cubic feet in a piece of marble weighing 5100 pounds. 

5. If sand weighs 120 lb. per cubic foot, find the number of 
cubic feet in a load of sand weighing 4800 pounds. 

6. A train of 12 sleeping cars is 840 ft. long. If the cars are 
all the same length, how long is each car? 

7. A steel rail 30 ft. long weighs 720 lb. ; what is its weight 
per yard of length? 

8. An iron beam 24 ft. long weighs 1080 lb. ; what is the 
weight of a piece of the beam 1 ft. long? 

9. A steel girder (or beam) weighs 1728 lb. Each foot of 
length weighs 48 lb. How long is it? 

10. There were 487,918 foreign immigrants to the United 
States in the year 1901. What was the average number per 
month? per day? (30 da. = 1 month.) 



DIVISION 



65 



11. During 1900 there were 448,572 immigrants. Find the 
average number per month. 

12. Answer the same question for 1891, 1892, 1893, 1895, 
and 1897, the numbers for these years being in succession 560,319; 
623,084; 502,917; 258,536, and 230,832, 

13. In Albany, N. Y., there are 30 mi. of street railway, 
operated by 600 men. What is the average number of employees 
per mile? 

14. From the data hei-e given answer the same question for 
these cities : 



City 



Miles 



EMPLOYEES 



St. Joseph, Mo. . 
Memphis, Tenn. 
Oakland, Cal... 
Hartford, Conn. 
Worcester, Mass 
Peoria, 111 



35 


175 


70 


490 


80 


560 


33 


660 


43 


473 


50 


275 



15. Find the value of x in each case: 

(1) 3264+ 6 = a: 

(2) 9432-^12 = 0; 

(3) 2247 -^ 7 = a; 

16. How can you prove the correctness of your work in divi- 
sion? 

17. Find what x equals in these equations: 



(4) 89,765 -^ 5 = a; 


(7) 24,568 f- 8 = a; 


(5) 78,870 -^ 11 = a; 


(8) 45,960 -h 12 = a: 


(6) 75,699-^ 9 = .r 


(9) 1,241,196-^11= a; 



(.)?-.o 


^ ^ X 


(5, *f = 49 


(7) 


«f- 


^ ^ X 


(«) 7 = 90 


(«) 



640 

X 

990 

X 



= 20 



66 



§51. Long Diyision. 

When dividend and divisor are hoth large numbers, it becomes 
necessarj to show all the steps of the work. 



06 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



1. 14,487-^33 = ? 



30 
400) 


= 439, quotient 


33)14487 
13200 


= 400 X 33 


1287 
990 


= 30X33 


297 
297 


= 9X33 


Shorter Form 




439, quotient 


divisor, 83)14487. dividend 
132 




128 
99 




297 
297 



Solution. —Beginning at the left of the 
dividend; 88 is not oontained in 1, nor in 
14, but it is contained in 14,400, 400 times. 
Write the 400 above the dividend and 
subtract 400 X 88 = 18, 200 from the divi- 
dend, leaving 1287. This remainder must 
also be divided by 88. 

88 is not oontained a whole number of 
times in 1, nor in 12, but it is contained 
30 times in 1280. Write the 80 above the 
400 over the dividend and subtract 80 X 88 
= 990 from 1287, leaving 297. 

88 is contained in 297, 9 times, leaving 
no remainder. 

Thus we see 83 is contained in 14,487 
400 + 80 + 9 = 489 times. 

The work may be shortened a little by 
omitting the zeros and writing numbers in 
the shorter form below. 



Check: 489 X 33 = 14,487. 
division is probably correct. 



Since this is the given dividend, the 



When a zero appears in the quotient proceed as follows : 
2. 18,722^40 = ? 



407 

46)T8722 

184 

322 
322 



Solution. — Begin as above. 46 is contained in 187, 4 times, 
with the remainder 8. Bring down the 2 in the dividend, 
giving 32. 46 is not contained in 82 a whole number of times. 
Write in tens place in the quotient and brin^ down the 
next 2 of the dividend, giving 822. 46 is contained in 822, 
7 times. Write the 7 in units place in the quotient. 



Check: 407 X 46 = 18,722, which equals the dividend. 



Complete these equations and check your work : 

3. 3,580 -^ 45= 7. 33,768 h- 72 = 

4. 15,552-4-64= 8. 35,096-^82 = 
6. 18,144^56= 9. 62,328^-84 = 
6. 20,083-^72= 10. 44,928^-96 = 



DIVISION 07 

§52. Exercises. 

1. There are 52 wk. in a year. My friend is 1872 wk. old; 
how many years old is he? 

2. There are 160 sq. rd. in 1 A. and a farmer pays 75^ per acre 
for cutting and binding wheat. How much will it cost to cut and 
bind the wheat on a field 68 rd. by 80 rods? 

3. A farmer paid a man $21.00 to shock his wheat, wages 
being $1.50 per day. How many days did the man work? 

4. In 1880, 25 farm wagons sold for $2250 and in 1900, 15 
snch wagons sold for $855. How much less was the average cost 
of a farm wagon in 1900 than in 1880? 

5. In 1880 a Minnesota farmer paid $3900 for 12 twine bind- 
ers and in 1900, 14 twine binders cost him $1680. How much had 
the average price of twine binders fallen during tl\ese 20 years? 

C. A steel rail weighing 72 lb. per yard is 30 ft. long. How 
many naen are needed to carry it, each man carrying 90 pounds? 

7. In 25 da. a man earned $56.25 husking corn at 3^ per 
bushel. How many bushels per day did he husk? 

8. A city lot 175 ft. long, containing 8750 sq. ft., sold for 
$6000. If the short side fronts the street what was the price per 
foot of frontage? 

9. The force required to draw a street car on a level track is 
35 lb. per T. (2000 lb.) of the combined (total) weight of the car and 
its load. What force is needed to draw a car weighing 5600 lb. 
when it is loaded with 60 passengers whose average weight is 140 
pounds? 

10. At the speed of an ordinary horse car the pull of a horse 
equals about 125 lb. of force in drawing the car. How many 
horses will be needed to draw the car of problem 9, no horse to 
draw more than 125 pounds? 

11. At a slow walk a horse can exert about 330 lb. of force. 
The force required to draw a loaded wagon on a level pavement 
is TfV of the weight of the wagon and load. A coal wagon 
weighing 4860 lb. is loaded with 4 T. of coal. How many horses 
will be needed to draw the load over a level pavement, no horse 
drawing more than 330 pounds? 



(J8 



NATIONAL 6RAMMA11 fiCflOOL ARtTHMETlC 



12. A horse can exert 1540 lb. of force for a few minutes. 
A box car weighing 30 T. is loaded with 32 T. The force needed 
to move the loaded car is ^ of the combined weight of the car 
and load. How many horses will be needed to start the car on 
a level track? 



§53. Larger Numbers. 

1. 5,128,672-1-9272 = ? 



653 

9272)5128672 
46360 

49267 
46360 

29072 
27816 

1256 



Solution. —The divisor, 9272, being too large to use 
readily, we first use a trial divisor. For the same 
reason, we select a trial dividend. 92, the trial divisor, 
is contained in 612, 5 times; but as the whole divisor con- 
tains four digits the partial dividend must be enlarged. 
9272 is contained in 61,286, 5 times. The quotient fi^re 5 
is of the same order as the last figure of the trial dividend, 
wiiich is hundreds. We write 5 in hundreds place in the 
quotient. Multiplying the whole divisor hj the quotient 
ngure we have the product 46,860. Subtracting this prod- 
uct from the trial dividend, 4926 remains. 4926 hundreds 
= 49,260 tens ; and 49,260 tens + 7 tens = 49,267 tens. Con- 
tinue in the same manner with each step that follows. 
The last subtraction gives a remainder of 1256. This remainder must 
also be divided by the divisor 9272. 

5,128,672 -«- 9272 = 553J}f f 

9272 
553 

27816 
46360 
46860 

5127416 
1256 

5128672 



The 553 is the whole, or integral part of the quotient and 
the mi is the fractional part. 



Check: Multiply the divisor bv the quotient, and to the 
product add the remainder. The result should equal the 
dividend. 



Solve the following problems and check your work: 

2. 131,320-^530= 4. 630,861 + 2731" 

3. 195,930 + 624= 5. 1,057,536 + 4352 « 



§54. Geography. 

1. From the table of §23 the area of Massachusetts is seen to 
be 8315 sq. mi., and that of Illinois is 50,050 sq. mi. How many 
states the size of Massachusetts could be made from Illinois? 



DIVISION ^9 

2. From the same table the area of Xew England is found to 
be 66,465 sq. mi. How many states as large as New England 
could be made from Texas? 

3. Make and solve other problems like these, using the table. 

4. The same table shows the area of Connecticut to be 4990 
sq. mi. and its population for 1900. to be 908,420. How many 
persons per square mile are there in Connecticut? 

Note. — In problems such as this, where the fractional part of the 

Quotient has no meaning, drop the remainder if it is less than half the 
ivisor, and add one unit to the whole part of the quotient if the remain- 
der is more than half of the divisor. 

5. The table of §28 gives the population of Connecticut for 
1890 as 746,258. What was the population of Connecticut per 
square mile in 1890? . 

6. Answer questions 4 and 5 for your own state. 

7. From the same table answer questions 4 and 5 for Okla- 
homa territory. 

8. Make and solve similar problems for any states you are 
studying in your geography. 

9. The area of Switzerland is 15,781 sq. mi. and its popula- 
tion is 2,933,334. What is the population of Switzerland per 
square mile? 

10. 640 A. = 1 sq. mi. How many square miles in 534,528 
acres? 

11. The area of the state of Texas is 265,780 sq. mi. ; that of 
New Jersey is 7815 sq. mi. How many states the size of New 
Jersey could be made from Texas? How many the size of Delaware, 
Avhich contains 2050 square miles? 

12. Porto Kfco has an area of 3531 sq. mi., and a population 
of 953,243. How many inhabitants does it support to the square 
mile? 

13. The greatest ocean depth found is 31,614 ft. near the 
island of Guam, in the Pacific ocean. The highest mountain in 
the world is Mt. Everest in Asia, which rises 29,002 ft. above sea 
level. Find the difference of level in miles between the greatest 
ocean depth and the greatest land altitude. 



70 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



14. The state of New York, with an area of 49,170 sq. mi., 
supports a population of 7,208,894. How many inhabitants does 
it average to the square mile? 

15. Hawaii has an area of 6449 sq. mi., and a population of 
154,001. Find the average population per square mile. 

16. In 1891 the total wheat area of North and South Dakota 
was 4,882,157 A. ; the yield was 81,819,000 bu. What was the 
average yield per acre? 

17. In the year 1900 Kentucky had 22,488 A. of rye under 
cultivation. The total yield was 292,344 bushels. "What was the 
average yield per acre? 

18. In the year 1890 there were 210,366 persons employed in 
manufacturing in Chicago. The total wages paid amounted to 
$123,955,001. What was the average wage paid to each person? 

19. In 1880, 3519 factories in Chicago together yielded 
$249,022,948 worth of products. In 1890, 9977 factories yielded 
$577,234,446 worth. During which year was the average product 
per factory greater, and by how much? 

20. A comparison of the density of population (population per 
square mile) may be obtained by finding the density of population 
of these countries: 



Country 


AREA 


Population 


Density 


German Empire 

Great Britain 


208,880 
120,979 
115,903 
204,092 
110,646 
8,660,395 
3,688.110 


56,345,014 
41,454,578 
26,107,304 
38,641,333 
82,449,754 
185,000,000 
76,212,168 




Austria 


France 


Italy 


Russia 


United States 





Note. — Only the whole numbers need be found for these quotients. 

§55. Diyision by Multiples of 10.— oral work 

1. Multiply each of these numbers by 10: 

568 1268 306 186.50 $8.65 

2. Multiply each of the same numbers by 100 ; by 1000. 

3. Divide each of these numbers mentally by 10: 

5680 12680 3060 $865.00 $86.50 



DIVI8I0K 71 

4- Make a rule for dividing any number quickly by 10; by 100; 
by 1000; by 1 with any number of zeros after it. 

5. Name these quotients orally : 

60 -^ 10 = ? 6 -^ 2 = ? 60 ^ 20 = ? 

860 ^ 10 = ? 86 ^ 2 = ? 860 -k 20 = ? 

$64.20 H- 10 = ? $6.42-^2 = ? $64.20-^20 = ? 

Examining your answers to the questions just asked, make a 
rule for dividing a number quickly by 20 ; by 200 ; by 2000. 

6. 180 H- 10 = ? 18^3 = ? 180 + 30 = ? 
630 -i- 10 = ? 63 ^ 3 = ? 630 + 30 = V 

From these answers make a rule for dividing any number 
quickly by 30; by 300; by 3000. 

7. Make a rule for dividing a number quickly by 40; by 400; 
by 4000; by 50; by 800; by 1200; by 1500. 

8. Make a rule for dividing any number quickly by any whole 
number of tens, as 40, 70, 90, 160; by any whole number of hun- 
dreds; of thousands. 

9. Cutting off zero from the right of a number has what effect 
^n the number? cutting off 2 zeros? 3 zeros? 

66 -i- 10 = 6«o. or 6.6. 
165 -^ 10 = 16/o, orl6.5. 

75-^-100=/o"o. or.75. 
478 -f- 100 = 4/o8o» or 4.78. 

10. Using first 10 and then 100 as a divisor, give and show the 
quotients of the following : 

400 500 2200 3300 
460 790 4280 4860 
287 439 9647 5732 

11. There are 10 pk. in a barrel. How many barrels in 1488 
pecks? 

12. There are 60 lb. in a bushel of potatoes. How many bushels 
in 486 pounds? 

13. 100 lb.= 1 cwt. How many hundredweight in 825 pounds? 

14. 200 lb. pork = 1 bbl. How many barrels in 7624 pounds? 



72 BATIOKAL GRAMMAR SCHOOL ARITHMETIC 

16. How many minutes in 4260 seconds? 

Solution.— When there are ciphers at the right of both n^ 

dividend and divisor, cut off an equal number of ciphers *y 

from both and divide. 6J*)4260 

IG. How many barrels will be needed for 4800 lb. of beef, 
allowing 200 lb. to the barrel? 

§56. Other Methods of Shortening Diviiion. 

1. Solve these'problems: 

5000)25000 500)2500 50)250 6)25 

How do the quotients compare? the dividends?^the divisors? 

2. Remembering that the dividends stand above the line and 
the divisors below, solve these problems : 

324^^ 108^, 36^^ 12^ 

81 • 27 • 9^ • 3 

How do the quotients compare? the dividends? the divisors? 

3. How do the quotients, the dividends, and the divisors com- 
pare in these problems : 

256^.^ 64^ 16^^ 4^ 

128 • 32 8 * 2 

4. To divide 324 by 81 what smaller numbers may I use to get 

the same quotient? How can I obtain these smaller numbers from 

324 and 81? 

Solution.— We see that, as 824 = 27x13 and 81=27x3, we may 
write 

12 X 27 _ 12 _ 

8 X 27 ~ 3 ~ 
This can be indicated thus : 

12x3?^ 



= 4. 



3xaT" 

Removing these factors is called caiicellation. It can often 
be used to simplify the division of products. 

5. Answer the same questions for 256 -^ 128. 
These problems show that any factor of both dividend and 
divisor may be dropped from bothand the remaining factors divided. 



DIVISION 73 

6. Solve these problems by cancellation : 

... 34x16 ^ 6x8x3 ,, 15x21x846 ,, 

^^ 2x16"" ^^2x8x3 * ^^^ 3x 7x846 * 

18 x3x4x67 _, ,5)625 ,,. 324 _ , ,^. 85 _ , 

^^ 2x4x3x67 • ^^25 ^^89"' ^^^34"* 

To use cancellation effectively, methods of finding factors of 
numbers are necessary. 

§57. Tests of Divisibility. 

1. Which of these numbers are exactly divisible (can be exactly 
divided) by 2 : 

12 24 36 23 45 18 37 40 59 61 

What are the last digits of the numbers which 2 will divide? 

Test for the factor 2: If a number ends in 0, 2, 4, 6, or 8, it can be 
exactly divided by 2. 

2. Which of these numbers are exactly divisible by 10: 

24 30 45. 50 700 640 83 765 6400 , 
What is the last digit of the numbers which 10 exactly 
divides? 

Test for the factor 10: If a number ends in 0,_10 exactly divides it. 

3. Make a rule for testing whether 100 divides a number. 

4. Which of these numbers does 5 exactly divide : 

16 18 35 25 60 20 28 65 460 675 1260 

What are the last digits of the numbers which 5 will divide? 

Test for the factor 5: If a number ends in or 5, 5 exactly 
divides it. 

5. Which of these numbers does 3 exactly divide: 

12 17 24 81 27 93 64 75 126 324 185 

Of the numbers which 3 exactly divides, will 3 also exactly divide 

the sum of the digits? Of the numbers 3 does not exactly divide, 

is the sum of the digits exactly divisible by 3? 

Test for the factor 3 : If 3 exactly divides the sum of the digits 
of a number it divides the number also. 

6. Of these numbers what ones does 9 exactly divide: 

126 368 453 729 819 639 2358 



74 BATIONAL GKAMMAR SCHOOL ARITHMETIC 

See whether 9 will exactly divide the sum of the digits of 

the numbers that 9 exactly divides. 

Test for the factor 9 : If 9 exactly divides the sum of the digits 
of a number it also exactly divides the number. 

7. Which of these numbers does 4 exactly divide: 

113 124 368 560 375 486 1204 

See whether of the numbers it exactly divides 4 also exactly 

divides the number indicated by the last two digits. For 

example, in the third, 368 can be exactly divided by 4 and so also 

can 68. 

Test for the factor 4 : If the number denoted by the last two 
digits of a number can be divided by 4, the entire number can be exactly 
divided by 4. 

8. Which of these numbers are divisible by 25 : 

60 175 285 625 1350 1275 8645 8675 8625 8650 

25 divides 175 exactly and 25 also exactly divides 75, which is the 

number denoted by its last two digits. Is this true of all numbers 

25 exactly divides? Is it true of any numbers 25 does not exactly 

divide? 

Test for the factor 25: If the number denoted by the last 2 digits 
of any number is divisible by 25 the entire number is divisible by 25. 

Test for the factor 6: test for both 2 and 3. 

Test for any composite factor: test singly for ail the factors of 
the composite factor. 

9. Test for divisibility by such numbers as 36, 216, 27, 
49, etc., which contain some factor two or more times. 

10. Pick out the numbers of this list which are exactly 
divisible by 2 : 

6 81 65 72 86 129 9864 8643 7986 16,835 29,860 

11. Pick out those which can be exactly divided by 3; by 9; 
by 4; by 5; by 10; by 12; by 15. 

§58. Checking Division. 

Division may be checked by multiplying the divisor by the 
quotient and adding the remainder to the product. If the sum 
equals the dividend, the work is checked. 



DIVISION 75 

Another check is to divide by the factors of the divisor suc- 
cessively and note whether the final quotient is the same as that 
given by the complete divisor. 

To check by casting out the nines, add the excess in the 
product of the excesses of divisor and quotient to the excess of the 
remainder. If the excess of this sum equals the excess of the 
dividend, the division is probably correct. 

Illustration. — Check the work of problem 1, §53. 

Cast the 9's out of the dividend, 5,128,672. The excess is i. 

Cast the 9's out of the divisor, 9272. The excess is 2. 

Cast the 9's out of the quotient, 553. The excess is 4. 

Cast the 9's out of the remainder, 1256. The excess is 5. 

The product of the excesses of divisor and quotient is 8. 

The excess of 8 is <9 itself. Add this 8 to the excess of the remainder, 
giving 13. 

The excess of this 13 is 4 and as this equals the excess 4 of the diyjr 
dend, the division is probably correct. 

It is even more important in division than in multiplication 
to examine a problem carefully before beginning to solve it. Try 
to foresee about what the answer must be. This often avoids 
blunders. 

Illustration. —1. If it requires 480 slates to cover a square (100 sq. ft) 
of roof surface, how many squares are there in a roof which requires 
13,680 slates to cover it? 

Pupil should at once notice that if it required 500 slates to cover a 
square, there would be a little more than 13,680 -«- 500, or 136 -4- 5 = 27 
squares and he might guess 28 squares. The aotual division of 13,680 by 
480 gives 28.5 squares. 

First form a rough estimate of the answer and then solve these 
exercises : 

2. $238 was paid for flour @ $3.50; how many bbl. were 
bought? 

Suggestion. — How many bbl. would there have been if the price had 
been $7.00 a barrel? 

3. In still air a hawk flew 375 miles in 2J hr. What was its 
speed per hour? 

4. In 3.9 hr. a crow flew 97.5 mi.; find the rate of flight per 
hour. 

5. From a certain cow ^ of the milk was butter-fat. The 
cow gave 12 lb. of milk per day. How many pounds of butter-fat 
will the milk from this cow yield in 70 days? 



76 



KATIONAL GRAMMAR SCHOOL ARITHMETIC 



§59. Town Block and Lots.— Fig. 20 is a drawing, or scale map, of a 
town block. Scale: 1 in. equals 100 ft. If possible, pupils should 
measure and draw to scale a block or field in the neighborhood 
of the schoolhouse, and in all problems use the numbers they 
obtain from their own measurements in preference to those given 
in the exercises. 




> 
< 



>- 
z 

Ld 

X 



B 



D 



^^i^^^:^^^^^^:^^:^}:^, !^,^^ 



l li^^i^^^^i^^i:^^^:^^^^^ 




t 



MARKET 



ST. 



—1 I — I — I 



i in. divided into tenths 



j^^^.vss.v^^s^s.^v<.^^^^^^ 



Figure 20 

Measure the drawing in Fig. 20, and find how long the block 

is between the sidewalks; how wide. 

Note. — Use the scale of tenths given in the figure. Lay a strip of 
paper having a straight edge beside the marked inch and mark short 
tines on the strip to indicate the tenths. 



DIVISION 77 

WRITTEN WORK 

1. How many square feet in the area of the block mnpo^ 

2. At $20 per ft. of frontage on (distance along) Market street, 
what is lot L worth? 

3. Make problems like 2 for other lots on Market street, using 
price per front foot of lots where you live. 

4. At $12.50 per ft. of frontage on Mathews avenue, what is 
lot H worth? 

5. Make similar problems for other lots on Mathews avenue. 

6. At $18 per ft. of frontage on Race street, what is lot G 
worth? Similar problems should be made by the pupil. 

7. Make problems for any, or all, of the lots, at the price per 
front foot where you live. 

Note. — Each property holder is taxed to provide funds for foundation 
material, brick, and labor to pave in front of his property to the middle 
line of the street. This is called an assessment. 



8. The streets are to be paved with brick. The cost of exca- 
vating (digging out) to the proper depth is 30^ per square yard of 
surface;* find the cost of excavating a strip 1 yd. wide extending 
from outside edge of sidewalk to middle of Race street. Ans, 13.00. 

9. What would be the cost of excavating a similar strip 1 ft. 
wide? 5 yd. wide? a strip as wide as frontage of lot E? Ans, $1.00; 
$15.00; $50.00. 

10. The cost of foundation material is 36^ per square yard; 
find the cost of enough such material for the strip 3 ft. x 30 ft. 
described in problem 8. Ans, $3.60. 

11. Find the cost of foundation material for each of the three 
strips (1 ft. X 30 ft., 15 ft. x 30 ft. , and 50 ft. x 30 ft.) of problem 9? 

12. The cost of the brick to be used is $10 per M. (thousand) 
and 78 bricks are needed to cover 1 sq. yd. What is the cost of 
the brick needed for the strip of problem 8? for each of the three 
strips of problem 9? 1st Ans, $7.80. 

13. What is the total cost of excavating, of foundation mate- 
rial, and of brick for the strip of problem 8? for each of the three 
strips of problem 9 (1 ft. x 30 ft., 15 ft. x 30 ft., and 50 ft. x 30 ft.)? 

♦Use prices current in your community whenever they can be obtained, instead of 
prices given. 



78 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

14. Find this total for other lots fronting on either Market 
or Race street. 

15. The labor of construction (making) costs 75^ per sq. yd. 
What is the cost of labor on a strip of the size mentioned in 
problem 8? Arts. $7.50. 

16. What will be the assessment against lot F for paving for 
each foot of frontage? 

17. Make similar problems for other lots, not including the 
corner lots. 

18. Henry and Mathews avenues, which are 30 ft. wide 
between sidewalks, are to be paved in the same way as Market 
and Race streets. Rates being the same as above, what will be the 
total assessment against lot L? (Omit the street crossings.) 

19. Make and solve problems like 18 for other lots. 

20. The owner builds a house on lot L. The length of the 
house is 36 ft. and the width is 30 ft. How many square feet of 
the lot are covered by the house? 

21. The southeast corner of the house is located 30 ft. from 
the south and east lines of the lot. Locate the three other 
corners. 

22. Find the cost, at 45^ per sq. ft., of the concrete walk 3 
ft. wide and 76 ft. long on lot L, as shown in the drawing. 

§60. House and Furnishingpi. — oral work 

1. What is meant by a l-brick wall?* a 2-brick wall? 

2. If it takes 7 bricks to make 1 sq. ft. of wall surface when 
laid in a l-brick wall, how many bricks per square foot are needed 

for a 2-brick wall? a 3-brick wall? a 5- 
brick wall? 

3. From the dotted lines in Fig. 21, 
can you tell how measurements may be 
taken on a wall to avoid counting comers 
twice? 











t 

Inside Measure 

i 











Outside-Measure 

FIGUKB21 ^' -^ brick is 2 in. by 4 in. by 8 in. ; 

how many cubic inches are there in it? 

"A l-brlck wall is a wall one brick thick, bricks lying on the largest surfaces. 



DIVISION 



79 



5. How many square inches in one end? one edge? one of its 
largest sarfaces? 

6. In locating the foundation of the house on lot L (Fig. 20), 
in what direction would you wish the line of the front foundation 
wall to run with reference to the Market street line? 

7. Point oat some of the square comers on the plans in 
Pig. 22. 

8. Can you point out any corners that are not square? 



' WRITTEN WORK 

The owner of lot L, Avhich fronts on Market street, builds the 
house whose foundation plans are given in Fig. 22 and whose first 
and second floor plans are given in Fig. 23. 



I T 1: N 

^^.. ..1; w4-: 



1 






*— --ioJi * 


^---11^ > 



FOUNDATION 



Figure 22 



1. What will it cost to excavate a rectangle 21' x 36' to a depth 
of 6 ft., for the foundations and cellar, at 20^ per cubic yard? 

Think of the foundation as made up of straight walls such as 
are shown in the second part of Fig. 22. The mark (') means foot 
or feet, and (") means inch or inches. 

2. The foundation is inclosed by 2 side walls, each 36' long, 
and 2 end walls, each 18^' long. Point out these walls in both parts 
of Fig. 22. The foundation walls are all 8' high. Find the area 
of the north surface of the north side wall. 

Ans. 288 sq. ft. 

3. What other wall has an outer surface equal to the surface 
mentioned in problem 2? 



80 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

4. Find the area of the outer surface of the east end wall. 

A?is. 148 sq. ft. 

5. What other outside surface equals this one in area? 

6. Masons reckon that it takes 14 bricks for each square foot 
of outer surface to lay a solid 2-brick wall. If all walls are 
2-brick walls, and solid, how many bricks will be needed for the 
north foundation wall? for the east wall? the south? the west? 

1st Arts. 4032; 2d Ans. 2072. 

7. How many bricks will be needed for all four of the outside 
walls? 

8. There are two inside foundation walls each 18^ ft. long, also 
one 10| ft. long, and one 11 ft. long. All are 2-brick walls 8 ft, 
high. If these walls contain no openings, how many bricks will be 
needed for the inside foundation walls? Afis. 6552. 

9. The outside walls contain 8 window openings each 1^ ft. by 
3 ft. If masons allow for one-half the area of all openings in 
computing (finding) the number of bricks, how many bricks should 
be deducted (subtracted) for the outside walls? Ans, 252. 

10. The inside walls contain 4 door openings each 2^ ft. by 8 
ft. How many bricks should be deducted for the inside walls? 

Ans. 560. 

11. Find the total number of bricks needed for the foundation 
walls and their cost at $9 per M.* 

12. If hauling costs 75^ per load of 1^ T. and each brick weighs 
6 lb., find the cost of hauling the brick for the foundations.! 

13. If the mortar costs $1.25 per M. bricks, what will be the 
cost of the mortar? (Use 18 M.) 

14. Four brick piers 2 ft. by 2 ft. and 4 ft. high support the 
porch columns. How much will the bricks needed for these col- 
umns cost at $12 per M., counting 22^ bricks for each cubic 
foot? (Use here the nearest half thousand, i.e. 1^ M.) 

15. While the house was building the owner decided to replace 
weather-boarding by stained shingles on a belt running around 
the house (which is 30' x 36') and extending to a distance of 9 ft. 
below the eaves. Each square yard, thus changed, cost $1.75 

♦ In computing the cost of brick use the nearest whole thousand. 

t When the last load is fractional, the price for a full load is charged. 



DIVISION 



81 



extra, no allowance being made for openings. How much doea 
this change add to the cost of the house? 

16. Hardwood floors were decided upon later to take the place 
of pine floors in the dining-room and front hall. This increased 
the price by 12^ per sq. ft. How much did this add to the cost of 
the house, counting the dining-room full 14' 6" wide and making no 
allowance for fireplace in hall. Ans, $51.54. 




FIRST FLOOR 



SECOND FLOOR 



Figubess 



17. A room is said to be well lighted when the floor area is not 
more than 6 times the area of the window surface which admits 
light. If all windows as shown in the plan are 2^ ft. wide and 6 
ft. high, and \ of the window space is covered by the sash, is the 
dining-room well lighted? is the kitchen? the hall? 

18. Is your schoolroom well lighted? Are the halls of your 
schoolhouse well lighted? 

19. A hall 6 ft. by 8 ft. is lighted by a pane of glass 2^ ft. by 
4 ft. in the upper'part of a door. Is the hall well lighted? 

20. The assembly room of a church is 40 ft. by 60 ft. and is 
lighted by 18 windows, each containing 24 panes of glass, 1 ft. by 
1^ ft., and by a large front window containing 40 such panes. Is 
the room well lighted? 

21. A room 12 ft. by 15 ft. has 3 windows, 2 containing 2 
panes of glass each, tte panes being 1^ ft. by 3 ft., and 1 coi> 
taining 2 panes of glass 3 ft. by 4^ ft. The room also has a door 
2i ft. by 7 ft. which admits light when open. Is the room well 
lighted if the window blinds are up and the door is open? Is it well 
lighted if the blinds are up and the door closed? If the blinds 
are half-way up and the door open? If the blinds are J of the way 
up and the door closed? 



82 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



§61. Applications of Cancellation. 

In all the problems of this list indicate your divisions and 
multiplications, then apply the tests of divisibility, and cancel 
the factors found in both dividend and divisor. Multiply the 
uncanceled factors in the dividend together and divide this product 
by the product of the uncanceled factors of the divisor. 

Illustration. — A speed of 102 mi. per hour equals how many feet 

per second? Indicate the work thus: 

34 22 

102X528^21 mx»6 34X22 748 ,.^. 
- = — = = — - = 149|. 



60X6^. 



5 



Ans. 149| ft. per second. 

1. It took 225,280 lb. of steel rails to lay 1 mi. of single-track 
railroad. What was the average weight per yard of the rails? 

2. Find the number of cubic 

feet in a ton of 2000 lb. for each 
of the substances given in the 
table. 

3. The average speed of 
American express trains is about 
35 mi. per hour for long dis- 
tances. IIow many feet per sec- 
ond is this? 

Note.— 60 sec. = 1 min. ; 60 rain. = 1 hour. 
Solution. — Put work in this form: 
7 22 
35 X 52813 ^X»9 154 





Lb. per 
Cu. FT. 


Cu. Ft. 

PERT. 


Granite 

Limestone.. 

Marble 

Sandstone . . 

Slate 

Cast Iron . . . 
Steel 


170 
160 
170 
140 
170 
450 
480 





60X613 



3 



= -75- = 511. -^^*- '^li ^t. per second. 



4. In both England and America, the average speed of express 
trains for distances from 100 to 250 mi. is about 40 mi. per hour. 
How many feet per second is this? 

5. For long distances the average speed of English express 
trains is about 43 mi. per hour. How many feet per second is 
this? 

6. In 1893 an express train in the United States ran 1 mi. at 
the rate of 98 ml. per hour. How many feet per second is this? 



DIVISION 83 

7. A railroads train ran for 1 min. at the speed of 130 mi. per 
hour. Find the number of feet per second which it traveled. 

8. Sound t]»vels in air at the rate of about 750 mi. per hour. 
How many feet per second is this? 

9. Gannon balls have been thrown at a speed of 810 mi. per 
hour for a few seconds. This is how many feet per second? 

10. The moon moves around the earth at a speed of about 
1,840,000 mi. in 30 da. How many feet is this per second? 

Note.— 24 hr. = 1 day. 

11. In 365 da. the earth moves around the sun through a 
distance of about 584,000,000 mi. What is the earth's speed in 
miles per second? 

12. The earth, by turning on its axis, carries a place on 
its equator about 25,000 mi. in 24 hours. How many feet is it 
carried per second? 

13. A farmer bought 18 bu. of wheat at 75^ and paid the bill 
with potatoes at 60^ per bushel. How many bushels of potatoes 
were required? 

14. A farmer bought 80 acres of land at $100 per acre and paid 
for it with land worth $60 per acre. How many acres were 
required? 

15. A man buys 5 acres of city land at $720 per acre and pays 
for it with 40 acres of farm land. What price per acre does the 
farm land bring? 

16. How many bricks each 2" x 4" x 8" are there in a straight 
pile 12" X 36" x 20'? In a straight pile 2' x 6' x 12'? 

17. How many lots each 25' x 150' can be made from a town 
block 250' X 300'? 

§62. Farm Products of the United States in 1900. 

In these problems use the tests for divisibility (§57) to discover 
common factors in divisor and dividend, and then remove these 
factory by cancellation. The numbers that remain in dividend 
and divisor are then to be divided in the usual way. Omit frac- 
tions in the quotients. 

1. There were 512 coffee farms in the United States in 1900, 



84 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

having a total area of about 70,200 acres. Find the average size 
of a coffee farm. 

Suggestion. — Test for the factor 8 in dividend and divisor. 

2. The total value of these 512 coffee farms was about $1,933,- 
000. What was the average value of a farm? 

3. If the total acreage (number of acres) of the coffee farms 
was about 70,200, and their total value about $1,933,000, what 
was the average value of an acre of coffee land in the United 
States? 

Suggestion. — Test first for the factor 100 and then for 2. 

4. There were 441 farms in the United States devoted to rais- 
ing a certain food plant, having a total acreage of about 18,900. 
Find the average size of one of these farms. 

Suggestion. — Test first for 9, then remove the common factor 
7 from both dividend and divisor. 

5. The total value of these 441 farms was about $562,500. 
Find the average value of one of these farms. 

6. If the total acreage in these farms was 18,900 and the total 
value $562,500, find the average value of an acre for raising this 
plant. 

7. There were in our country about 5,720 rice farms, having a 
total acreage of about 108,800 and a total value of about $17,834,- 
000. Find the average acreage and the average value of a rice 
farm, also the average value of an acre for raising rice. 

8. About 2,025 nursery farms had a total acreage of about 
165,800 and a total value of $19,146,000. What were the average 
acreage, the average value of a nursery farm and the average value 
of land per acre for nursery purposes? 

0. For sugar farms these numbers were nearly correct : 7,340 
farms, of 2,669,000 acres, worth $150,420,000. Solve problems 
like 7 and 8 for sugar farms. 

10. The numbers for farms devoted to raising flowers and 
plants were about 6,160 farms, of 42,660 acres, worth $52,460,000. 
Sulve problems like 7 and 8 for flower- and plant-raising farms. 



DIVISION 



85 



11. From these numbers, solve like problems for fruit farms : 
number of farms 82,200; total acreage, 6,150,000; total value, 
$440,000,000. 

12. Further problems may be made on this table, which is a 
fairly complete list of the farm products of the country. 



Pboduct 


Number or 
Farms 


ACRES TS FARMS 


Value of Farms 






Total 


Average 


Total Averag 


Tobacco 


106.000 


9,570,000 




$ 215,500,000 


Vegetables . . . 


186.000 


10,160,000 




547,000,000 


Dairy produce 


. 358,000 


43,300,000 




1.693,500,000 


Cotton 


1,072.000 


89.600.000 




1,108,000,000 


Hay and grain 


1,320,000 


210,260,000 




6,380.000,000 


Live stock 


1,565.000 


355,000,000 




7,505,300.000 


Miscellaneous. . 


1,060,000 


113,200,000 




2,383,700,000 



§63. The Thermometer. — The official record of hourly temper- 
atures for Chicago from 6 p.m. February 27 to G p.m. March 1, 
1902, as given in a daily paper, was as follows: 



Feb. 27; Night 


Feb. 28; 


Day 


Feb. 28 ; Night 


MA30H 1; 


Day 


6 p. m 

7 p. m 


.38*' 
.38 
.37 
.37 
.39 
.39 
.38 

.38 
.38 
.38 
.38 
.37 
.39 
.40 


6 a. m. . . 

7 a. m. . . 

8 a. m. . . 

9 a. m. . . 

10 a. m. . . 

11 a. m. . . 

12 m 

Average. . 
12 m 


. . . .40^ 
....41 
....41 
....41 
....43 
....44 
....38 

.' .* .' .38 


6 p. m 

7 p. m 

8 p. m 

9 p. m 

10 p. m 

11 p. m 

12 midnight 
1 a. m 

' 2 a. m 

3 a. m 

4 a. m 

5 a. m 

6 a. m 

Sum 

Average ... 


.39" 
.38 
.37 
.37 
.37 
.37 
.36 
.36 
.35 
.34 
.34 
.34 
.34 


6 a. m. . . 

7 a. m. . . 

8 a. m. . . 

9 a. m. . . 

10 a. m. . . 

11 a. m. . . 

12 m 

1 p. m. . . 

2 p. m. . . 

3 p. m. . . 

4 p. m. . . 

5 p. m. . . 

6 p. m. . . 

Sum 

Average. . 


....3r 

....34 
....33 
....33 
....32 
....32 
. . . .32 
....31 
....31 
....30 
....29 
....28 
....28 


8 p. m 

9 p. m 

10 p. m 


11 p. m 

12 midnight 

Average 

12 midnight 

la. m 

2 a. m 


1 p. m. . . 

2 p. m. . . 

3 p. m. . . 

4 p. m. . . 

5 p. m. . . 

6 p. m. . . 
Average . 


....35 
....35 
....37 
....39 
....40 
....39 


3 a. m 

4 a. m 

5 a. m 

6 a. m 

Average 



Note. — The average of temperatures, or numbers, is found by adding 
them and dividing the sum by the number of temperatures, or numbers, 
which have been added. To average means to find the average. 



86 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Feb. 27 from 6 



p.m. 



to 



1. Average the temperatures on 
midnight; from midnight to 6 a.m. 

2. What was the average temperature on Feb. 28 
from 6 a.m. to 12 m.? from 12 m. to 6 p.m.? 

3. What is the difference between the two aver- 
ages of problem 1? of problem 2? What do these 
differences mean? 

4. What is the difference between the lowest and 
highest temperatures from 6 p.m. Feb. 27 to 6 p.m. 
Feb. 28? This is called the range of temperature 
for the day. 

5. The average temperature for Feb. for 30 yr. is 
26°. How much does the average temperature for 
the 24 hr. following 6 p.m. Feb. 27 exceed the thirty- 
year average? 

6. Find the average temperature for the night 
of Feb. 28; for the day of March 1. 

7. Find the difference between these averages. 
What does this difference show? 

8. Make and solve such problems from outdoor 
thermometer readings at your schoolhouse. 

9. What is the reading of the thermometer of Fig. 24? What 
would the thermometer read if the top of the mercury column 
stood at "freezing"? at ^'summer heat"? at "blood heat"? 

10. Average the thermometer readings taken at your own 
school for each hour from 9 a.m. to 4 p.m. for a month, and keep 
in a notebook a record of your readings and averages. 

11. Lay off 
on cross -lined 
paper the read- 
ings of problem 
10 as the read- 
ings of the table 
from 9 a.m. to 4 
p.m. of Feb. 28 
are laid off in Fig. 25 ; draw a smooth curve, free-hand, through 
He points. Such a curve gives a picture of temperature changes. 





mik 


^1 


a 1 


% ■ 


1}^ 




IID- 




ITD- 




toa^ 


KT 


'bo- 




\T. 


iC 


70- 

eo- 

S0-: 




% 




♦tJ- 




i3D- 


^ 






IQ- 




\^-. 




iO: 




10^ 




£0: 




|30-= 




♦a- 




^1 



Figure 24 



50' 
40" 
30' 
20' 
10' 
01 







::n. 


"1 








11 














^^ 


















■» 














—J 


:- 















9 10 11 12 1 2 3 4 

A.M. A.M. A.M. M. P.M. P.M. P.M. P.M. 

February 28 
Figure 25 



10 11 12 I 2 3 4 

P.M. P.M. P.M. M, A.M. A.M.A.M.A.M. 

Febrttarii 28— March l 
Figure 26 



DIVISION 87 

12. From a daily paper obtain the hourly readings from 9 p.m. 
to 4 a.m. and lay them off to scale as the readings of the table 
from 9 p.m. Feb. 28 to 4 a.m. March 1 are laid off in Fig. 26. 
Draw the free-hand curve as before and find whether the day or 
night temperatures are the steadier (more regular) . 

13. Average your readings from 9 a.m. to 4 p.m. and draw a 
straight, horizontal line on your diagram, at a distance above the 
horizontal zero line equal to the average, as the line av of Fig. 25 
is drawn. This is the average line for these 8 hours. 

EXERCISES FOR PRACTICE 

Check results in division by multiplying divisor by quotient 
and adding the remainder if there is one. 

SHORT AND LONG DIVISION 

1. Divide 1260 by 2; by 3; by 6; by 5; by 7; by 8; by 10. 

2. Divide 2880 by 3; by 4; by 5; by 6; by 9; by 10; by 12. 

3. Divide 3402 by 3; by 7; by 21; by 4; by 12; by 28. 

4. Divide 2520 by 4; by 6; by 20; by 6; by 7; by 30; by 8. 

5. Divide 33264 by 6; by 7; by 8; by 42; by 56; by 48; by 9. 
G. Divide 50400 by 10; by 30; by 40; by 50; by 60; by 90. 

7. Divide 201600 by 200; by 300; by 400; by 500; by 600; 
by 700; by 900. 

8. Divide 3360 by 7; by 8; by 56. 

9. Divide 2880 by 8; by 9; by 72. 

10. Divide 57672 by 6; by 2; by 12; by 72. 

11. How may you divide a number by 35 by short division? by 
56? by 63? by 80? by 72? by 54? by 48? by 84? by 96? 

LONG DIVISION 

12. Divide 360360 by 120; by 210; by 336; by 416; by 1248. 

13. Divide 5045040 by 42; by 105; by 315; by 540; by 702; 
by 862;. by 1404. 

14. Divide6387624by 81; by 320; by 576; by 736; by 826. 

15. Divide 8736824 by 632; by 82G1; by 42125; by 143216. 

16. Divide 30200610 by 700; by 1764; by 1908; by 36821. 

17. Divide 23642901 by 21 ; by 86 ; by 364; by 4081 ; by 19881 ; 
by 98631. 

18. Divide 38083584 by 672; by 816; by 2688; by 8064. 



«8 



RATION^AL GRAMMAR SCHOOL ARITHMETIO 



BILLS AND ACCOTTKTS 



g64. Exercises. 





MON. 


TXJ. 


Wbd. 


TH. 
HOURS 


FRI. 


Sat. 


Wagbs 


HOURS 


HOUBS 


HOURS 


HOURS 


HOURS 


S 


CTS. 


AdftniR 


8 


8 


SH 


9 


su 








Benson ....,,. 


8 


8H 


SH 


SH 


S}i 


9 








Boyd .... 





4 
8M 


6 


6 


6H^ 


7 








Olaussen 


8 


SH 


S}i 


9 


8M 








DenninfiT 


SH 


9 


8>i 


S}i 


9 


9 




*"^**" "o 


Doan 


8 


8 


8 


8 


8 


8 









FiGURR 27 

1. Find the number of hours each man worked. 

2. Find the number of hours all the men worked. 

3. At 30^ an hour, find the daily wages of each man. 

4. Find the amount due each man at the close of the week. 

5. Find the amount due to the six men at the close of the week. 

6. Which man worked the greatest number of hours? the 
smallest number? 

7. Counting 9 hr. to the day, what is the greatest amount 
any one man could earn in a week? 

8. Counting 8 hr. to the day, how much "overtime" did each 
man work? 

9. For how many extra hours did five of the men work, count- 
ing 8 hours to the day? 

10. If double wages were paid for ''overtime," how much did 
each man earn during the week? How much did all earn? 

11. How many hours did the third man lose? 



§65. 



ORIGINAL PROBLEMS 



1. Adams paid $4 on a doctor's bill, and put aside $2 for rent. 

2. Benson has a wife and two children. Average amount 
for each one in the family. 



BILLS AND ACCOUNTS 



8i) 



3. How mucli does Boyd's loss of time reduce his wages for 
the week? 

4. Claussen paid $10 for clothing, and $3.50 for the week's 
board. 

5. Denning saves $3 a week to pay the premium (yearly 
charge) on a life insurance policy, pays $3.50 a week for board, 
and 35^ for laundry. 

§66. Accounts. 

A man has on hand at the beginning of the month of February, 
1902, $250. He receives during the month $200 for salary and 
$60 for the rent of two houses. During the month he pays $72.50 
on an insurance policy, $110 for household expenses, $25.50 for 
incidentals, and $5 for books and stationery. His account book 
shows the following entries : 
Dr, {Receipts) Cash {Expenditures) Cr» 



1902 
Feb. 1 

** 15 
**27 



On Hand . 

To Salary. 

"Rent .. 



28 On Hand 297 00 



$250 

200 

60 



510 



00 



1902 
Feb. 1 



By Insurance 

** Household Exp. 
** Incidental " 

" Books and Stat'y 



Balance . 



$72 

110 

25 

5 

297 



510 



00 



1. With what items is cash the man's debtor? With what items 
is cash his creditor? 

2. During the month, how much money did the'man receive? 
How much money did he actually possess between February 1 
and 28? 

3. How much was spent? What is the difference between receipts 
and expenditures? How much more was spent than was received 
as salary? 

Draw forms and prepare cash accounts for the foUovring: 

4. March, 1902. On hand first of the month $1000. March 
5, sold land for $1500; received on the 15th, |l50 for rent; and 
sold a team for $225 on the 25th. Bought real estate on the 8th 
for $1000; paid $65.25 for repairs on the 15th; taxes on the 16th 



90 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



amounted to $17.50; household expenses for the month amounted 
to $100, and personal expenses to $50. 

5. June, 1902. A boy's receipts and expenditures were as fol- 
lows: Received 25^ for mowing the lawn on the 1st; 50^ a day for 
running errands on the 4th and 5th; 25^ for mowing the lawn 
again on the 15th. On the 17th he paid $1.25 for a bat. The 
same day he earned 50^ for delivering packages for the grocer. 

6. July, 1902. A farmer sold on the 13th two loads of hay at 
$20 a load; 4 hogs weighing in all 1000 lb. at 6^; 15 lb. butter 
at 20^. On the 13th he bought 26 lb. granulJ^ted sugar at 6^; 
3 lb. coffee at 35^ ; and dry goods to the amount of $15. On the 
26th he sold vegetables to the amount of $1.25, and 9 doz. eggs at 
10^. The same day he bought clothing to the amount of $10. 

7. Find cost of each purchase, also the amount due, on the bill 
below : 



E. S. WiCKWIRE, 

In account with Mannheimer Bros., 



Dr. 



1902 
Jan. 4 
- 4 
•' 12 
*' 14 


To 1 Rue at S75.50 










** 1 Overcoat *' 35.00 


•* 7 yd. Dress Goods ** 2.00 

'•3M'' Silk - 1.50 

Amoiint'diie 

















Mannheimer Bros. 



Received payment, 
Jan. 31, 1902. 

Draw forms and prepare similar bills: 

8. Edward Morris in account with J. W. Bowlby, Dr., St. Paul, 
Minn., May 30, 1902. 1 doz. collars at 25^ a piece; i doz. pr. 
cuflfs at 25^ a pair; 2 neckties at 50ji5; 3 shirts at $1.25; 1 hat at 
$3.50; 1 pr. shoes at $3.50; 1 pr. gloves at $1.25. Paid, May 30. 

9. A. H. Simons bought of Yerxa Bros., Chicago, June 6, 
1902, 1 sack appleblossom flour at $1.30; 8 lb. ham at 17^; 15 
lb. granulated sugar at 6^; i lb. oolong tea at $1.60; 3 lb. Mocha 
and Java coffee at 40^ ; 1 bunch of celery at 30^. Paid in full the 
same day. 



BILLS AKD ACCOUNTS 



91 



10. Edward Ryan bought of M. J. Doran, February 7/1902, 
2 cords of hard wood at $8.75; 3 T. of hard coal at $8.25; 2 
cd. of pine slabs at $3.25. 

11. May 4, 1902, Alvin Johnson bought of the John Martin 
Lumber Co., cash payment, 1000 cedar shingles at $3 per M; 1 
bbl. of lime at 68^; 1000 ft. of pine flooring at 65^ per M; 3000 
laths at $1.50; 15 lb. shingle nails at 7^. 

Accounts are not always settled in full at the close of the 
month. When a portion of the amount due is carried over to the 
next month, this item appears as the first item on the bill, under 
the title "accounts rendered." 

12. Find amount of this. bill: 

J. FmESTONE, 





To Allen, Moon & Co. 


. Dr. 








1902 
Feb. 1 

- 10 
" 10 
'' 10 

- 10 
" 15 
•' 15 


To Acc't rendered 






24 


85 


'• 2 Sacks of Flour at $2.60 

" 25 tb. Ham 18 






'' 4 doz. Eggs... 22 

** 61b. Butter 25 

" 25 lb. Granulated Sugar. . . .06 
'* Vegetables 5.00 

By Cash 













13. A. L. Parker, in account with Marshall Field, lacks $12.75 
of paying the entire bill of February, 1902. His March purchases 
were as follows: March 5, J doz. towels, @ 50^ a piece; 2 table 
cloths, @ $6.50; 1 doz. napkins, @ $5.50 a doz. ; 2 bed spreads 
@ $2.50; ) doz. sheets @ 75^ a piece; J doz. pillow cases @ 20^ 
a piece; 7 yd. dress goods @ $1.50; findings, $5.25. Paid in full 
March 28. 

14. Edward D. Young in account with P. W. Salisbury, wood 
and coal dealer, has left oyer from a previous account $8.25. On 
the 3d of January, 1902, he bought 2 T. of hard coal @ $8.25; 1 
cd. of hard maple wood, $8.50; 1 cd. of pine slabs, $3.50. Paid 
the whole amount January 29. 



92 RATIONAL GBAMMAB SCHOOL ARITHMETIO 

15. On Nov. 15, I purchased 1 can lima beans® 12^; 1 can 
peas @ 12^; 1 qt. cranberries @ 12^; 4 lb. pork chops @ 12^; 
4 J lb. chicken @ 12^ ; 2 lb. butter @ 32^, and paid the account 
with a $5.00 bill. What change lyas due me? 

§67. Problems of the Grocery Clerk.— A grocery boy made the 
following transactions (sales and purchases), including the filling 
of the orders, and returned the correct change without an error all 
in one hour. What change did he return to each customer? 

1. The first customer bought 

Celery $0.05 Potatoes $0.15 

Eggs 22 Sugar. .10 lb. @ 4^^ 

and paid the clerk with a $1.00 bill. 

2. The second customer bought 

Salt $0.10 Apples fO.25 

Flour 55 Breakfast food 13 

Yeast 02 Eice. .3 lb. @ 8^^. 

Milk 07 

and he paid with $1.50. 

3. The third customer bought 

Tomatoes $0.20 Beans. . i . . .4 qt. @ 7^. 

Soap 25 Tea 2 lb. @ 60^. 

Butter. .3 lb. @ 27^. Cauliflower. .3 heads @ 18^. 

Coffee.. 3 lb. @ 33 J j^. 
and paid with a $5.00 bill. 

4. The fourth customer bought 

Eggs $0.22 Lampwicks $0.10 

Bacon 18 Cheese 16 

Peas 13 

and paid with a $5.00 bill. 

5. The fifth customer bought 

Commeal $0.25 Melon $0.25 

Sweet potatoes 27 Pineapple 20 

Candy 10 Lard. .3 lb. @ 13i^. 

Grapes 18 

and paid with a $2 bill. 



BILLS AKD ACCOUNTS 



93 



6. The sixth customer was a farmer, who sold to the grocer 
Eggs 6 doz. @ 18^. Tomatoes. . .2 bu. @ 75^. 



Butter 15 1b. @ 22^. 

Potatoes. . . .5 bu. @ 35^. 
and bought of the grocer 

Sugar 20 1b. @ ^^. 

Coffee 31b. @ 25^. 

Cheese 2 lb. @ 18f 

How much money should the grocer pay the farmer to balance thb 
account? 

7. A grocer's daily sales for 4 weeks are given below. Without 
rewriting the numbers find the total sales (a) for each week ; (b) 
for the 4 weeks ; (c) the average daily sales for the month. 

Wbbkly 
Totals 



Sugar corn. .6 doz. @ 10^. 



Tea 2 1b. @ 55^. 

Flour 1 sack, $1. 15 

Rice 6 lb. @ Sif 



MON. 


TUBS. 


Wkd. 


THtTB. 


FBI. 


Sat. 


$28.75 


$37.25 


$35.18 


$68.12 


$20.13 


$86.58 


18.62 


9.68 


21.83 


40.28 


37.60 


75.76 


30.18 


29.95 


23.61 


9.61 


10.84 


68.94 


19.27 


39<13 


28.16 


38.19_ 


5.63 


98.56 



§68. Family Expense Account. 

Foot the following problems as rapidly as you can work accu- 
rately. 

1. The household expenditures (money spent) of a family are 
here given for each of the first 7 da. of Oct., 1900. Find the daily 
expenditures and the total expenditure for the week. 

Oct. 1. Oct. 2. Oct. 3. 
Oatmeal. $0.11 Laundry $0.48 Bread $0.15 

Sweet potatoes .18 

Apples 25 

Bread 10 

Meat .20 

Sundries G3 

Bread 20 

Crackers 15 



Meat 18 

Peaches 25 

Grapes 13 



Dried beef. 

Tea 

Bread 

Celery . . . . 
Soap 



.10 
.30 
.20 
.05 
.25 



Total . 



Total. 



Baking powder 

Cheese 

Macaroni 

Bananas 

Honey 

Peanuts 

Tomatoes .... 
Shoe polish . . 

Carfare 10 

Onions .10 

Total.... 



.25 

.10 
.15 
.15 
.20 
.05 
.15 
.10 



9G SATIOKAL OBAMMAB SCHOOL ABITHMETIO 

Datb Apr Mat Junb Jult Aug. 

1 $ 52.86 $ 53.48 $ 56.75 $ 56.73 

2 3.87 2.50 .67 1.28 

3 3.68 2.18 1.20 2.10 

4 6.36 1.28 .85 3.50 

5 1.86 .25 .75 4.05 

6 2.15 8.12 3.80 1.10 

7 75 3.62 2.37 .05 

8 1.95 .67 2.12 .12 

9..... , 3.28 4.10 .55 .15 

10 2.18 .10 .96 .35 

11 4.10 1.01 2.84 3.60 

12 2.15 .86 3.69 .10 

13 1.00 1.52 2.50 2.18 

14 10 .78 .55 .00 

15 1.61 2.58 1.86 1.57 

16 3.21 3.65 1.68 .76 

17 57 .25 2.75 .56 

18 3.76 2.51 .99 3.65 

19 15 .86 6.24 .05 

20 4.25 1.76 3.26 8.29 

21 86 2.08 .25 .00 

22 95 1.15 4.86 .00 

23 87 3.12 .28 1.68 

24 1.16 1.28 .83 1.00 

25 3.15 3.75 1.96 2.28 

26 1.82 .15 1.76 .05 

27 2.78 1.67 3.75 6.28 

28 60 2.18 1.28 .00 

29 3.25 1.01 8.14 3.16 

30 96 3.87 1.00 2.15 

31 00 .75 .00 1.58 



56.10 


t .65 


2.50 


54.10 


6.82 


4.06 


.04 


.60 


.00 


2.76 


1.28 


.26 


.76 


1.52 


4.26 


.68 


1.18 


2.12 


2.60 


.39 


.00 


1.86 


3.10 


3.24 


.69 


3.27 


1.58 


1.68 


3.82 


.66 


.66 


1.68 


6.63 


2.17 


.28 


.69 


.12 


3.00 


.17 


.12 


2.16 


6.81 


2.60 


.06 


.05 


.10 


8.16 


7.06 


.16 


.19 


.08 


2.60 


1.20 


2.97 


2.00 


3.60 


3.60 


.10 


.25 


2.65 


1.68 


.00 



Monthly I 
Totals ) 

Total 

6. Find the daily average per month for each of the 6 months. 

7. Find the monthly average for 6 months from April to 
September. 

8. Find the monthly average for the whole year. 

9. Find by adding horizontally the total expenditure for the 
first day of these 6 months ; for the second day. 



BILLS AND ACCOtJNM 



9? 




Figure 2S 



§69. The Equation. 

1. y lb. of sugar are placed on the left scale pan, and a weight 
of 10 lb. on the right pan balances it (see Fig. 28). How 
heavy is y? 

The balance of these two weights 
is expressed thus: y = 10. (I) 

This expression is called an equa- 
tion^ and is read *'y equals 10." 

2. If a 4 lb. weight is now added 
to the right pan, how many additional 
pounds of sugar must be placed upon 
the left pan to balance the scales? 
Write an equation to express the relation between the weights 
now in the pans. 

3. If X lb. on the right balance y lb. on the left, how would 
you state the fact in an equation? 

4. If 20 lb. are added to the x lb. already on the right,-how 
many pounds must be added on the left to balance the equation? 

5. If 35 is added on the right side of equation (I), what 
change must be made on the left side to balance the equation? 
Write the equation thus changed. 

Just as the horizontal position of the scale beam shows that 
there is a balance of the weights on the pans, so the equality sign, 
= , shows that there is a balance of value of the numbers between 
which it stands. It must never be used between two numbers, 
or combinations of numbers, that are not equal in value. 

The number on the left of the sign of equality is called the first 
member^ or the left side of the equation. The number on the right 
is called the second fnember, or the right side of the equation. 

6. Four equal weights and a 5 lb. weight just balance 21 lb, 
(Fig. 28.). How heavy is one of the equal weights? 

Solution.— 4a?+ 6 = 21. Subtract 6 from both sides 
5 5 



4a; = 16. If 4a? = 16 what is x1 What is the answer 
to the problem? 

7. 3y+2=14; findy. 8. eoa; -f 12 = 32 ; find a;. 9. 7rr-f8 = 
29;finda;. 10. 9;2;4- 3 = 75; find ^. 11. 15a + 3 = 48; find a. 




98 KATIOKAL GRAMMAR fiOHOOL ARITHMETIC 



CONSTEXTCTIVE OEOMETET 

§70. Problems with Euler and Compass. 

. Problems I. to XI. are to be solved with tuIgt 
and compass. Keep all the pencil points sharp 
while drawing and work carefully. 

The simple instrument shown in Fig. 29 is a 
form of the compass which will do for these 
problems. 

The pencil point of the compass will be 
called the pencil foot, or pen foot. The other 
point will be called the pin foot. 

Problem I. — Draw a circle with i inch 
FiGXjBB 29 . radius. 

Explanation. —First Step : Place the pin foot on an inch mark of your 
foot rale and spread the compass feet apart until the pencil foot just 
reaches to the next half inch mark. 

Second Step: Without changing the distance 
between the compass feet, put the pin foot down at 
some point, as A, Fig. 30, of your paper and, with 
the pencil foot, draw a curve entirely round the 
point A. 

A curve drawn in this manner is called a circle. 
How far is it from A to any point of the curve? 

Any part of the whole circle, as the part from C 
to D, or from D to Bis an arc of the circle. 

The point A, where the pin foot stood, is the figure ao 

center. 

The distance straight across from B, through ^ to C, is a diameter. 
BC is a diameter. 

The distance from A straight out to the circle is a radivs. The 
plural is radii (ra'-di-i). AC, AB and AD are all radii. 

What part of the diameter equals the radius? 

EXERCISES 

1. Draw circles with these radii: 

r; 1"; ir; ir; ir; 2". 

2. Draw circles with the same center A, and with these radii: 

r; r; r; 1; U". 

Circles whose centers are all at the same point are called con- 
centric circles. 

Problem II. — Draw (or lay off) a line equal to a given line. 




CONSTRUCTIVE GEOMETEY 90 

Explanation.— Let the given line AB (Fig. 31) have the length a 
units. 

With nder and pencil draw any straight line, as CX, longer than a. 

Place the pin foot on A and spread the v 

compass feet until the pencil foot just ^ a g 

reaches to B. Without changing the dis- \j) 

tance between the feet, put the pm foot on C j J[ 

C and with the pencil foot draw a short Figuhbsi 

arc across the line CX as at D. 

Then CD is the desired line; for if we call its length a?, we have made 
x=a. 

EXERCISES 

1. Draw lines equal in length to these given lines: 
a b c 



2. Draw lines having these lengths : 
1"; If; 2r; 21"; 3^'. 

Problem III. — Draw a line equal to the sum of two or 
more given lines. 

Explanation.— Let the two given lines be a and b. Fig. 82. First step : 
Draw the indefinite line CX longer than the combined length of a and o, 

and make CD equal to a as in 

? Problem II. 

/> Second Step: Spread the com- 

pass feet apart as far as the length 

V of b. Then put the pin foot on 

the crossing point (intersection) 

of the arc and line at D and draw 

v^nnnv^) auother short arc at E. How long 

^'^"""^ '' is DEI How long is CE^ 

CE is the desired sum. If we call its length 8, we may write this 
equation : 

(L) s = a-fb. 

With ruler and compass how can you find a line equal to the sum of 
3 given lines? of 4? of any number of given lines? This is called con- 
structing the sum of lines. 

EXERCISES 

1. Construct the sum of the lines in each column, denote the 
constructed line by 5, and write an equation like (I.) for eacli 
case: 

cam a 

den h 

e p m 



100 BATIOKAL GRAMMAR SCHOOL ARITHMETIC 

2. Construct these sums: 

i"+r+r+r; 2"4.j"+{"+}"; j^+sf+r+r- 

Problem IV. — Draw a line equal to the difference of two 

given lines. 

Explanation. — Let the two given lines be a and h. First step: Draw 
the indefinite line CX longer than the minuend line a. As above, make 

the minuend CD, equal 

S to a. 

b Second Step: Spread 

"T — . J. the compass points apart, 

r^ M^ W jg" as before, as far as the 

— Tf"'^ length of the subtra- 

Figure 83 hend line h. This is 

called '* taking b as a 
radius." Place the pin foot on D, swing the pencil foot back toward 
C, and draw the arc at E across the line CX. This makes DE how 
long? What line now equals the difference d between a and &? The 
equation for this case is: 

(XL) d = a'K 

EXERCISE 

Construct the differences of these pairs of lines, call each dif- 
ference d^ point it out in the construction and write the equation 
for each case : 

m c f CL 



Problem V. — Draw a line equal to 2, 3, or 4 times a given line. 

Explanation. — Let a be the given line. We are simply to draw 
a line equal to the sum 

of a and a. (See Prob- . a 

lemin.) ^ „ 



We may write C r 

(III.) p = 2a. '■ vi^ 

How would you con- P 



struct 3a? 4a? 5a? 7a? FigukeSI 

SarV 8i/? 6«? 

Write and read the equation for each construction. 

EXERCISE 

Construct three times these lines and give their equations 
(like III., above): 

a c d %a 3m 

Problem VI, — Divide a given line into two equal parts. 



CONBTKUCTIVB GEOMETRY 



101 



Definition. — Dividing a line into two equal parts is called bisecting 
the line. 

Explanation.— Let the given line be AB, Fig. 86, call its length a. 
First step: Spread the compass feet apart a little farther than 4 the 
length of AB, Place the pin foot 
^ " £3 



first on A and bring the pencil 
foot, by estimate, somewhere 
above the middle of the line a, 
and draw the arc 1. Then carry 
the pencil foot down, by estimate, 
below the middle of a and draw 
the arc 2. These arcs should both 
be drawn long enough to make 
sure that arc 1 passes over arc 2, 
under the middle point. 

Second step: Now change the 
pin foot to B, and without chang- 
ing the distance between the feet, 
draw arc 3 cutting arc 1 and 4 
cutting arc 2. Call the crossing 
points C and D. 

Third step: Place the edge of 
the ruler on C and D and draw 
the straight line CD, crossing a 
to I of a. Test by measuring i 

The equation for this case is 



A 



)^ 



Figure 85 

at E. Either AE or BE is equal 
Test by measuring with the compass. 



(IV.) g=la, org = ^. 

The first is read "g equals one-half a," meaning q equals i of a, and the 
second is read **g equals a divided by 2." Do they, therefore, mean 
the same thing? 



EXERCISES 

1. Draw a line J" long and bisect it. 

2, Bisect lines of these lengths: 

f'; ir; 3"; 5"; of; 3}". 





FlGUBB 86 



Figure 37 

3. Longer lines may be bisected at the blackboard with crayon 
and string (Fig. 36), or on the ground with a cord and a sharp 
stake (Fig. 37). 



102 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

Problem VII. — Construct an equilateral (equal sided) tri- 
angle with each side equal to a given line. 

Explanation.— Let a Fig. 38 be the given line. 

J Draw CX longer than a and 

make CD = a, as in Problem II. 

With length a between the 
compass feet, place the pin foot on 
C and draw arc 1, by estimate, 
above the middle of CD. 




Without changing the distance 

between the compass feet, place 

the pin foot on D and with the 

pencil foot cut arc 1 with arc 2. 

Call B the intersection (crossing 

FiGUBB 88 point) of arc 1 and arc 2. With the 

ruler draw a line from C to ^ and another from D to B. Then CDB is 

the desired equilateral triangle. 

EXERCISES 

1. Construct equilateral triangles having sides of these lengths: 
bed e 



2. Construct equilateral triangles having these sides: 

1"; 2"; 3"; 2J"; 4". 

3. With crayon and string construct these equilateral triangles 
on the blackboard : 

6"; 1'; U"; 18"; 24". 

4. With cord and nail or stake construct these equilateral 
triangles on the floor or ground: 

6'; 10'; 18'; 1 rod; 30'. _ 

Problem VIII.— Construct an isosceles (i-sds'-see-lees) trian- 
gle with the base and the two equal sides equal to given lines. 

Definition. — An isosceles tri- t 
angle has at least two sides equal. — — 

Explanation.— Let b, Fig. 39, . S 

equal the base, and e, one of tne 
equal sides. 

Make CD = h (Problem II). With 
C as center and radius equal to e 
draw arc 1. With the same radius 
and with D as center draw arc 2. 
Connect their intersection E with C 
and with D. ^ 

Then CDE is the desired isos- ^ 
oeles triangle; for we made CD = b, TTTrirBK « 

CE = e and DE= h, ^'^^"' ^ 

Definition.— The side, CD, which is not equal to either of the other 
two sides is the base. 




C0K8TRUCTIVE OEOMBTRY 



103 



EXERCISES 



1. The figures in Fig. 40 represent idl forms of the triangle. 
What is a triangle? 




Figure 40 

2. What triangle has all its sides equal? What is an equi- 
lateral triangle? 

3. What triangles have at least 2 sides equal? What is an 
isosceles triangle? 

4. What triangles have no two sides equal? What is a sca- 
lene (ska-leen') triangle? 

Problem IX. — Draw a scalene triangle with sides equal 
to given lines (no two heing equal). 



Explanation.— On the line CX 
make CD = a as in Problem I. 
With C as center and h as radius, 
draw arc 1. Then with D as center 
and with c as radius, draw arc 2 
across arc 1. Call their inter- 
section E. 

With the ruler draw line EC 
and ED. 

Then CDE is a scalene triangle Q. _ 

with the sides equal in length to 

a, 6 and c. figure 4 1 

EXERCISES 

1. Draw a scalene triangle having sides of these lengths: 
ah c 




2. Draw a scalene triangle having sides of 3", 4", and 5"; 
of 1", li", and 2". 

Problem X. — With the compass draw a three-lobed figure 
inside of a circle of \" radius. 



104 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



EXPLANATION.- 




-Draw a circle with V' radius around some point, as O, 
as a center. Set the pin foot at any point 
on the circle, as at 1, Fig. 42. and with- 
out changing the distance between the 
compass feet draw the arc from A on the 
circle through O to B on the circle. 

With same nulius and with pin foot 
on B, draw a short arc at fS. 

Put the pin' foot on 2 and with the 
same radius as before draw the arc BOCy 
C being on the circle first drawn. 

Put the pin foot on C and draw a short 
arc at 3. Then place the pin foot on 3 and 
draw an arc from C through O to A. 



Figure 42 




EXERCISES 

1. Draw a three-lobed figure using a radius of 1". 
^2.'^I)raw a three-lobed figure using as radius this line . 

3. Making the distance between 
the compass points J", and using the 
points 1, 2, 3, 4, 5, and 6 in turn, draw 
a six-lobed figure like Fig. 43. 

4. Color the lobes of your figure 
with a red lead pencil of with water 
colors and the spaces a, b^ c, d^ e, and 
/ with a green or yellow pencil or 
water colors. figure 4s 

Problem XI. — Draw a regular 6-sided figure (regular hexagon) 
within a circle of y radius. 

Explanation. — Draw a circle with 
center at O and with \'t radius. 

Starting at any point on the circle as 
at i, put the pin foot on 1 and draw the 
short arc B across the circle, keeping the 
radius J". Then put the pin foot on 2 and 
draw the arc 3. Then with pin foot on 3 
draw arc 4\ and so on around. 

How many steps do you find reach 
once round? 

Now with the ruler and pencil connect 
1 with 2, then 2 with 3, 3 with 4, and 
finally G with 1. 

The figure made by the 6 straight lines 
is the regular hexagon desired. 




Figure 44 



MEASUREMENT 105 



EXEBGISES 



1. Draw regular hexagons within circles of these radii : 1"; 2"; 

2. How long is one side of each of the hexagons drawn in 
exercise 1? How long is the sum of all the bounding lines of 
each hexagon? 

pEFiNmoN.— The sum of all the bounding lines of any figure is called 
the perimeter of the figure. 



MEASTTKEHEITT 
§71. Measuring Yalue. 

Money is the common measure of the value of all articles that 
are bought and sold. 

The unit on which United States money is based is the 
dollar. The dollar sign is $; thus 5 dollars is written $5 or $5.00. 
This unit is called the U. S. Standard of value. 

1. A dollar is worth as much as how many dimes? nickels? 
quarters? cents? 

To measure the yalue of one amount of money by another is 
to find how many times one of the amounts is as large as the 
other. 

2. Measure $1 by 50^; by 25^; by 20^; by 40^; by 75^. 

3. Measure $10.50 by 50^; by 75^; by $1.50; by $5.25. 

4. A farm is worth $1000 and a city lot $2000. Measure the 
yalue of the farm by the value of the lot. 

5. Measure the value of a $150 horse by the value of a $15 pig. 

6. Measure $75 by $5; by $25; by $15; by $150; by $225. 

7. Measure the value of 160 A. of land worth $75 per acre 
by $100; by $500; by $1000; by $8000; by $24,000. 

8. Which is worth the more, 80 A. of land @ $75, or 50 A. 
@ $112? how much more? 

9. Measure $400 by the value of an $8 calf; of a $25 colt. 



106 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

§72. Measuring Length and Distance. 

Answer these problems, first, by estimate, writing your esti- 
mates in a notebook ; then answer by actual measurement. Finally, 
compare your estimates with the results of your measurements 
and tell how much your estimates differ from the measurements. 

1. How wide is the page of this book? how long? How wide 
are the margins? 

2. How wide is a pane of glass in your schoolroom window? 
how long is it? 

3. How wide is the top of your desk? how long? How 
high is the top of your desk from the floor? How high is your 
seat? 

4. How high is the bottom of the blackboard from the floor? 
How wide is your blackboard? how long is it? 

5. How tall are you? How far can you reach by stretching 
both arms as far apart as possible? 

6. How far is it around your waist? How far is it around 
your chest, just below your arms, when as much of the air as 
possible is exhaled from your lungs? What is your chest 
measurement when as much air as possible is drawn into your 
lungs? 

7. The difference between these two chest measures is called 
your chest expansion. What is your chest expansion? How 
does your chest expansion compare with the average for the 
pupils of your room? 

Note. — ^You may easily increase your chest expansion by a little 
practice in deep breathing. 

8. How many steps wide is your schoolroom? how many steps 
long? 

9. How many feet long is your room? how many feet wide? 

10. How many yards long is it? how many yards wide? 

11. How many inches long is your step? how many feet long? 
how many yards long? 

12. How many inches long and wide is your schoolroom? 

13. How many feet long is your schoolhouse? how many 
yards long? 



MEASUREMENT 107 

14. How many steps wide and how many steps long is 
your school yard? how many feet wide and how many feet 
long? how many yards? 

15. How many steps is it from your home to the school- 
hbuse? how many feet? how many yards? how many rods (1 rd. = 
5i yd.)? 

16. How far can you walk in 1 min.? How many miles 
could you walk in 1 hr. at the same rate? 

17. How far is it from your home to a neighboring large city? 
Answer this by using a map and the scale given with it. How 
long would it take you to walk to that city at the rate of walking 
in problem 16? 

18. How long will it take a train to run from your nearest 
station to that city at 24 mi. per hour? 

19. Using the scale map of your state (see Geography) 
find the length and the width of your state; of your county; 
of the U. S. 

20. How long and how wide is a two-cent postage stamp? 

21. How long is the diameter (distance across) of a copper 
cent (see Fig. 30)? of a dime? of a nickel? of a quarter dollar? 
of a half dollar? of a silver dollar? 

22. How long is the radius of each of these coins? 

23. Wrap a strip of paper, or a string, around each of these 
coins and find how far it is around each. 

24. What kind of unit do you need to measure and express 
long distances? medium distances? very short distances? 

Note. — Such a number as three-eighths, or |, of an inch means three 
units each one-eighth of an inch long. Such a unit is called 9k fractional 
unit, and such numbers as i, {, {, \\, ^l, |f , are all said to be numbers 
expressed in fractional units. Name the unit of each of the six frac- 
tions just given. 

25. How long is the distance around {circumference) a 
bicycle wheel (wrap a string around the wheel)? How long is 
its diameter? 

26. Answer the same questions for a carriage wlieel ; for the 
bottom of a bottle; of a can; for the head of a barrel; for any 
other circles you can find. 



108 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

27. Arrange these measares thus : 



Objbct 


Diameter 




Silver dollar 






Half dollar 






Quarter dollar 






Nickel 






Dime 






Cent 






Bicycle wheel 






Carriage wheel 






Bottle 






Can 






Barrel head 













Fill out columns 2 and 3 of a table like this with your measurements 
and keep them for later use. 

























9 
























8 
























7 
























6 
























5 
























4. 


:k 


















5 


















2 


WLMJm 










1 ^ 



♦ 567 

Scale >V 



6 9 10 n IS 



§73. Measuring 8nr£Eu^s. 

A square unit is a square 1 unit 
long and 1 unit wide. 

1. The side wall of a room is 9 ft. 
by 12 ft. What will it cost to lath and 
plaster the wall at 3^ per square yard? 

2. Using your own measures of the 
side and the end walls of your school- 
room, answer the same question for 
its walls and ceiling. 

3. How many square feet of black- 
board surface are there in your room? 

4. How many square inches are there in a pane of glass in one 
of your windows? How many square feet of window surface 
admit light into your room? 

5. This should be at least as great as | of the floor surface of 
your room. Is it? 

6. ^ of an inch in the drawing, Fig. 46, stands for one foot 
in the room. Measure and find the number of square feet of 
plastered surface in the whole interior of the room, deducting 
the part covered by the baseboard and the floort 



Figure 45 



MEASUREMEKT 



109 



Such a representation of the room as that of Fig. 46, show- 
ing the walls and the ceiling spread out on a flat surface, is 
called a development of the room. 

7. Draw to scale, 
from your own meas- 
ures, the development 
of your own school- 
room. 

8. A schoolroom 
is 24' square and 15' 
high. How many 
sq. yd. of flooring are 
needed for the room? 
How many sq. yd. of 
plastering are needed 
for the ceiling? 

9. One side wall of 
the room (prob. 8) 
contains two windows, 
each 6' X 8' ; an end 
wall contains a window 
6i'x 8' ; and the other 
side wall has a window and a door, each 4'x 8'. How many sq. yd. 
of window and door surface in the entire room? 

10. The rest of the wall surface 
is plastered. How many square 
feet of plastering are there in the 
room? 

11. A barn roof is made up of 
two rectangles, each 30' x 40'. How 
many square feet are there in the 
roof? How many squares of roof- 
ing? (A square of roofing is a 
10-ft. square =100 sq. ft.). 

12. A house roof is made up of 
two rectangles, one 18' x 40' and 

the other 30' x 40'. How many squares of roofing in the whole 
roof? How many sq. yd. of roof? 





Ceiling 




End Wdl 


3ide Wall 


End Wall 










Base Board 
Floor 












BaS6 BoArd 
SideWall 





Scale ^'1 

FIGXTBB 46 




Figure 47 



110 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



13. How many acres (1 acre = 160 sq. rd.) are there in a field 
40 rd. X 80 rd.? 60 rd. x 80 rd.? 80 rd. x 80 rd.? 80 rd. x 160 rd.? 

14. What is the cost at $2.20 a sq. yd. of paving a street GO' 
wide and f of a mile long? 

15. What is the cost at 90^ per sq. ft. of making a cement 
sidewalk If yd. wide and f of a mile long? 

16. How many square feet of glass are there in a window con- 
taining 4 panes of glass, each 18" x 1 yard? 

17. A rug 9' X 12' is placed in a room 14' x 18', with its long 
edge parallel to the long side of the room and with its center at 
the middle of the floor. How many square feet of the floor remain 
uncovered? 

18. The streets of a city cut out a row of blocks like that of 
Fig. 48. The numbers along the sides are the lengths in feet. 
How many square feet are there in block -4? in J?? in (7? 




Definitions. — A four-sided figure, like A^ having square 
corners^ is called a rectangle, A four-sided figure, like G or Hy 
having two pairs of parallel sides^ whether the comers are square 
or not, is ^parallelogram. 




Figure 49 



19. What is the area of the rectangle of Fig. 49? 



MEASUREMENT 111 

20. Examine the parallelograms and the rectangles beneath 
them and find the areas of the parallelograms. 

21. If the length of any parallelogram and its distance square 
across (altitude) are given, how can we find a rectangle with the 
same area as the parallelogram? 

22. Point out in Fig. 48 a rectangle that has the same area as 
the parallelogram G. 

23. The length of a parallelogram is b ft. and its altitude is 
a ft. ; what is its area? 

24. Call P the area, h the length, and a the altitude of a par- 
allelogram; write an equation to show how you would use h and a 

^ to find P, 

\ 25. The streets, ST and LM, are parallel. Fig. 48. 
\ j How many square feet are there in block (r? in //? in 

\ I ^? ini^? 
52jI 26. Study Fig. 50 and find how many square" feet 
FIGUBB.50 ^YiQXQ are in D of Fig. 48. 

27. How can you find the area of a triangle when you know 
the length of its base and its altitude (shortest distance to the 
base from the opposite corner)? 

28. In the triangle, ABC (Fig. >^- yD 

51), call the area T, the altitude y/1 I \v y^ 

(OB) a, the base (AB), b. Write an ^ X I ^ ^ ^ 

equation to find T from a and b, ^ 

29. Find the area in square feet x figubb 5i 

of the following triangles of Fig. 48: Z?; C; D\ E\ F, 

30. The part of a roof in the shape of a parallelogram has a 
length (bane) of 18 ft. and a breadth {altitude) of 12 ft. What is 
the area of the parallelogram? 

31. A city lot has the shape of a parallelogram of 400 ft. base 
and 300 ft. altitude. What is its area? 

32. A field has the form of a triangle of G8 rd. base and 46 rd. 
altitude. What is its area? 

33. The triangular gable (see Fig. 10, p. 50) of a houso is 
24 ft. at the base and is 14 ft. high. What is tho area of the 
gable? 



112 



RATIONAL GBAMMAE SCHOOL ARITHMETIC 



34. Find the missing part (base, or altitude, or area) of the 



following parallelograms : 
Base 



Altitudb 
18 ft. 



Abba 



(1) 28 ft. 

(2) 40 rd 

(3) 

(4) 

(5) 

(6) 

(8) 
§74. Measuring Yolnme (Bulk) and Capacity. 

ORAL WORK 





40 rd. 


160 rd. 


eOJ rd. 
88 rd. 


m rd. 
p mi. 
X in. 


n rd. 
q mi. 



960 sq. rd. 
1600 sq. rd. 

28160 sq. rd. 



a; X y sq. m. 




Figure 52 




Figure 53 



1. How many cubic inches are there in a 2-inch 
cube? a 4-in. cube? an 8-in. cube? a 16-in. cube? 

2. How many cubic feet are there in a cubic yard? 
in a 6-ft.* cube? in a room 3 yd. x 3 yd. x 3 yd.? 

3. How many cubic inches are there in a 
5-in. cube? in a 10-in. cube? 

4. How many cubic inches are there in a 
9-in. cube? in an 18-in. cube? 

5. How many cubic inches are there in a 
cubic foot? 

WRITTEN WORK 



1. What is the capacity of a square cornered box 
3" X 4" X 6" (see Fig. 54)? 3' x 4' x 6'? of a room 3 yd. 
X 4 yd. X 5 yd.? a yd. x d yd. x c yards? 

2. A box-car 8' x 34' can be filled with wheat to 
a height of 5 ft. When full how many cubic feet of 
grain does it hold? 

3. If a bushel of wheat = | cu. ft., how many 
bushels does the car hold? 

4. Noticing that a cubic foot of grain (not ear corn) is ^ bu., 
make a rule for finding the number of bushels a wagon-box or 
a granary will hold when full. 




Figure 54 



HEASUBEHEKT 



113 




5. A wagon-box is 2' x 4' x 10'. How many bushels of ear-corn 
will it hold if f eu. ft. = 1 bu. of ear-corn? 

6. How many rectangular solids 4" x 3" x 7", will fill a box 
16" X 12" X 28 inches? (See Fig. 55.) 

7. A box 30" X 24" x 12" will contain 
how many blocks 6"x 4" x 3" inches? 

8. 128 cu. ft. = 1 cd. How many 
cords in a straight pile of wood 80 ft. 
X 4 ft. X 4 feet? :figubb55 

9. The volume of a rectangular bin is 1500 cu. ft. If it 18 
25 ft. long and 6 ft. deep, what is its width? 

10. How many bricks 8" x 4 " x 2" are there 
in a regular pile 218" x 24" x 48" inches? 

11. Measure the length, the width, and the 
height of your schoolroom, and find how many 
cubic feet of air it contains. 

12. How many pounds does the steel T-beam 
of Fig. 56 weigh if 1 cu. in. of steel weighs 
4i ounces? 

13. A steel I-beam 24' long has a cross section of the^form 
and size shown in Fig. 57. How much does it weigh 
if steel weighs 486 lb. per cubic foot (metal 1" thick)? 

14. A water tank is 3' x 6' x 10'. How many 
cubic feet of water does it hold when full? 

15. There are 231 cu. in. in a gallon; how many 
gallons does the tank of problem 14 hold? 

16. How many cubic inches of grain does a box hold if it 
is 8 in. square and 4| in. deep? how many gallons? 

17. How many cubic inches does a box hold if it is 16 in. 
square and 8^ in. deep? how many bushels (2150.2 cu. in. = 
1 bushel)? 

18. How many cubic inches does a box hold if it is 
9" X 10" X 12"? about what part of a bushel? 

19. Answer similar questions for a box 10 in. square and 
10} in. deep; for a box 10" x 12" x 18". 





Figure 57 



114 EATIOKAL GRAMMAR SCHOOL ARITHMETIC 

§75. Measuring Weight. 

Beview the oral work of §74. 

1. A 3-in. cube of soil weighs 20 ounces; how many pounds 
does 1 cu. ft. of the same soil weigh? How many pounds does a 
cubic yard weigh? 

2. A 4-in. cube of sand weighs 48 ounces ; how many pounds 
does 1 cu. ft. of the same sand weigh? How many pounds does a 
cubic yard weigh? 

3. A cubic foot of water weighs 1000 ounces ; how much does 
a 6-in. cube of water weigh? a 3-in. cube? 

4. A cubic foot of iron weighs 480 pounds; how many ounces 
does 1 cu. in. weigh? 

Suggestion. — Notice that 1 cu. ft. = 12 x 12 x 12 cu. in., then 
use cancellation, thus : 

10 

^0 4 

^^0 X 10 10x4 40 ,. . , • I, .4 

lUlUlt = T73 = "9 ^ ^♦^ ^^^- ^ ^"- ^^- ^'^^^^ ^* ^^• 
3 3 

5. A cubic foot of oak wood weighs 64 lb. ; how many ounces 
does 1 cu. in. of oak wood weigh? How many cubic inches of oak 
weigh just 1 pound? 

G. A cubic foot of copper weighs 552 lb. ; how many ounces 
does 1 cu. in. of copper weigh? How many cubic inches of copper 
weigh a pound? 

7. A cubic foot of gold weighs 1200 lb. ; how many ounces does 
1 cu. in. of gold weigh? How many cubic inches of gold weigh a 
pound? 

8. A cubic foot of zinc weighs 43G lb. ; how many ounces does 
1 cu. in. of zinc weigh? How many cubic inches of zinc weigh a 
pound? 

9. A bushel of coal weighs 80 lb., a short ton weighs 2000 lb., 
and a long ton weighs 2240 lb. How many bushels of coal are 
there in a car load of 36 short tons? of 36 long tons? 



MEA8UREMEKT 115 

10. Find the cost of 31 lb. of wire nails @ 4 cents. 

11. What is the cost of 1 lb. of sugar selling 22 lb. for a 
dollar? 

12. Postage on first-class mail matter is 2^ an ounce. What 
would the postage be on a 2f lb. package of first-class matter 
(16 oz. = 1 pound)? 

^13. Find the cost of 2f lb. butter @ 32 cents. 

14. New York merchants bought in 1 da. the following: 

C324 lb. butter at an average price of 22^ cents ; 
1988 lb. cheese at an average price of 14J cents; 
6840 lb. sugar at an average price of 4} cents; 
2780 lb. sugar at an average price of 4.8 cents. 

Find the total weight and the total cost. 

15. Find the total number of pounds and the total value of 
these purchases : 

650 lb. cut loaf sugar @ $5.74 per hundredweight (cwt.); 
825 lb. granulated sugar @ $4.80 per hundredweight; 
400 lb. powdered sugar @ $5.24 per hundredweight; 
700 lb. confectioner's sugar @ $4.89 per hundredweight; 
350 lb. extra white sugar @ $4.78 per hundredweight. 

16. Find the weight and total value of this shipment : 

5400 lb. plain beeves @ $5.70 per hundredweight; 

6375 lb. choice beeves @ $5.80 per hundredweight; 

8450 lb. fair beeves @ $4.90 per hundredweight; 

8625 lb. medium beeves @ $4.60 per hundredweight; 
40,700 lb. veal calves @ $6.50 per hundredweight; 
43,000 lb. western steers @ $8.50 per hundredweight; 

8450 lb. Texas steers @. $3.70 per hundredweight; 
25,200 lb. beef cows @ $2.85 per hundredweight. 

17. How many tons did the purchases of problem 14 weigh 
(2000 lb. = 1 T.)? How many tons in the shipment of problem 16? 

18. A troy ounce of pure gold is worth $20.67. How much 
is a troy pound of pure gold worth (12 troy oz. = 1 troy pound)? 



116 BATIONAL GRAMMAR SCHOOL ARITHMETIC 

19. An avoirdupois pound equals ly^j of a troy pound. 
About what is the value of an avoirdupois pound of gold? 

Note. — Add to the value of a troy pound of gold 31 times j}^ of its 
value. 

20. An avoirdupois pound = 16 avoirdupois oz. ; what is the 
value of an avoirdupois ounce of gold? 

21. What is the value of your weight in gold? (Your weight 
is given in avoirdupois pounds.) 

22. A grain dealer received during January 130 carloads of 
grain, averaging 25| T. each. Counting 60 lb. to the bushel, 
how jnany bushels did he receive during the month? 



§76. Measuring Temperature. 

On the Fahrenheit thermometer the point where water begins 
to freeze is marked 32°, and the point where water begins to boil 
is marked 212°. (See Fig. 24, p. 86.) 

1. How many degrees are there between the boiling point and 
the freezing point? 

2. At 11 p.m. on a. certain date a thermometer read 32°. 
The mercury then fell i° an hour for 4 hr. What was the 
reading at 3 a.m. the next day? 

3. The mercury then fell 1° an hour for 5 hr. What was 
the reading at 8 a.m.? 

4. At 3 a.m. on a certain day the reading was 26®. The 
mercury fell 2° an hour for 5 hr. What was the reading at 
8 a.m.? 

5. On a certain date the reading was 10°, and the mercury 
fell on the average lf° an hour for 7 hr. What was the reading 
at the end of the 7 hours? 

6. It then fell 1^° an hour for 8 hr. What was the reading 
then? 

7. The mercury fell from the reading 12° above zero to 3** 
below zero. How many degrees did it fall? 

8 We might write rea.dmg& above zero thus: A. 12°; A. 6**; 
A 32°; and readings below zero thus: B. 3°; B. 6°; B. SO*", 



MEASUREMENT 117 

How many degrees does the mercury fall from the first of these 
readings to the second? 

(1)A. 8° to A. 3°; (4) A. 5° to B. 7°; (7) B. 2° to B. 11°; 

(2) A. 8°toB. 3°; (5) A. 2° toB. 12°; (8) B. 7^° to B. 13? ; 

(3) A. 30° to B. 2°;. (6) A. 4^° to B. 4i°; (9) B. 4° to B. 12i°. 

9. How many degrees of rise or fall are there from the first of 
these readings to the second? If the change is a rise^ mark it R; 
if 9k f ally mark it F. : 

(1) B. 7° to B. 2°; (5) A. 9° to B. 9°; (9) A. B^to A. 12° 

(2) B. 2i° to B. 1° ; (6) A; 2*° to B. 2^° ; (10) A. 3° to B. 30^° 

(3) B. 2i° to A. 1°; (7) B. 15° to B. 6^°; (11) A. 18°toA.67i° 

(4) A. 3i° to B. 1° ; (8) B. 15° to A. 6^° ; (12)B. 22^° to A. 67^°. 

10. Instead of writing an A. for "above zero" readings, it is 
customary to use the sign (+). What sign would you then sug- 
gest for "below zero" readings? Tell whether a change from the 
first to the second of these readings denotes a rise or Bk fall in each 
case and by how much? 

(1) +16° to + 100°; (4) +32° to- 3°; (7) -18° to -34°; 

(2) + 32° to + 212°; (5) + 16° to - 17°; (8) - 18° to -6^°; 

(3) + 2° to + 161°; (6) + 8° to -30°; (9)- 6° to - 27^°. 

11. The 12 o'clock (noon) readings for 4 successive days were 
as follows : + 82^° ; + 78J° ; + 61^° ; + 534°. What is the average 
of these 12 o'clock readings. 

12. Find the average of these 9 a.m. readings for 6 da. : + 9° ; 
+ 4°; +5°; +12i°;+14i°; + 9i°. 

13. Find the average of these 6 readings: - 4°; - 6°; - 2°; 
-5°; -13°; -12°. 

14. Find the average of these 2 readings: -r 8° and -2^. 
Show on Fig. 24, p. 86, what point on the thermometer is midway 
between the readings +8° and - 2°. 

15. What point is midway between the readings + 13° and 
- 5°? between the readings + 4° and - 6°? + 12° and - 12"? 
-4° and -12°? 




118 RATIONAL GRAMMAR SCHOOL ARITHMBTIO 

§77. Measuring Time. 

The primary unit used in measuring time is the mean solar 
day. This day is the average time interval during which the 
rotation of the earth carries the meridian of a place eastward from 
the sun back around to the sun again. It is the average length 
of the interval from noon to the next noon. 

1. If the time piece is running correctly, how 
many times does the short hand (the hour hand) 
of a watch or clock turn completely around 
from noon to the next noon? 

2. How much time is measured by one com- 
plete turn of the hour hand? 

3. What part of a day of 24 hr. is measured 
by the rotation (turning) of the hour hand from 
XII to VI? from XII to III? from XII to I? to 

FIGURE 58 jj^ ^^ ^^^ 

4. What name is given to the time interval in which the hour 
hand moves from XII to I? 

5. For measuring shorter periods the motion of the long hand 
(the minute hand) is used. What part of a day is measured by 
the movement of the minute hand from XII around to XII 
again? from XII to VI? from XII to III? from XII to IX? 
from XII to I? 

6. What name is given to the interval of time required for the 
long hand to make one complete turn? 

7. How many times does the minute hand turn around while 
the hour hand turns around once? 

8. The small and rapidly moving hand covering the VI of the 
watch face is the second hand. How much time is measured by 
one whole turn of the second hand? 

9. Over how much space does the tip of the minute hand 
move while the second hand moves around once? 

10. The second hand circle is divided into how many equal 
parts? What name is given to the time interval in which the 
second hand moves over one of these equal spaces? 

11. How many times does the minute hand turn around in one 



MEASUREMENT 



119 



day? in a week? in a month of 30 da.? in 365 da. (1 common 
year)? 

12. Answer the same questions for the second hand. 

13. There are about 365^ da. in 1 yr. If the hour and the 
minute hands of a watch start at XII and the second hand at 60 
at the beginning of a year, how will the hands point at the instant 
the year ends, if the watch runs correctly and without stopping? 

14. Answer question 13 supposing the length of the year to 
be 365 da. 5 hr. 48 min. 46 seconds. 



w 



J Brown 


I 


■0 


3 




> 


tp 


• 




o 


I 






WHDugan 




!? 


f 




^ 


AS Park 




\ 


JM5mtth 



Section of Land— Figure 5» 



§78. Measuring Land. 

1. A section of land is a tract 320 
rd., or 1 mi., square. It is usually 
divided into halves, quarters, eighths 
and sixteenths, as shown. How many 
square rods make a section of land? 

2. An acre = 160 sq. rd. ; how 
many acres in a section of land? 
in a quarter section? 

3. A section of land is divided up into farms, as shown in the 
cut ; how many acres are there in the farm belonging to A. S. Park? 
to H. S. Barnes? to J. Brown? H. A. Dryer? J. M. Smith? P. S. 
Mosier? J. S. Doe? 

4. Point out the south half of the section; the east half; the 
Ni; theWi 

5. Point out the SE i; the NW i; the SW i; the NE \, 

6. Point out the N" J of the SE i; the E ^ of the SE \\ the 
W i of the NE i; the S i NW i; the W i E^. 

7. Point out the SW \ of the SE i; the NE \ SW i; the 
NWiNWi; theNE i SW i. 

8. The farm of E. Miner would be described as the N ^ E ^ 
E i NE \\ describe the farms of the 8 other owners of this section. 

9. How many rods of fence would be required to enclose the 
section and separate it into farms as shown? 

10. How many rods long is a ditch starting from the NE corner 
of H. A. Dryer's farm, thence running due south to H. S. Barnes's 



1^0 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



N line, thence due W to Barnes's W line, thence S to J. M. 
Smith's N line, thence due W across Park's farm to his W line? 
11. Each small square in Fig. 60 represents a section. The 
^ large square represents a townsJiip; how long 

is a township? how wide? 

12. How many square miles in 1 Tp.? how 
many acres? 

13. Point out in Fig. 60 the following: 

(1) NW i Sec. 33. 

(2) Si Sec. 17; NE^Sec. 17; EJNEi 
Sec. 17. 

(3) NW i Sec. 21; XE i NW i Sec. 21; SE i NW i 
Sec. 21. 

14. How many farms of IGO A. each could be made of a 
township? 

15. How long and bow wide is a square 160-acre farm? 



6 


5 


4 


3 


2 


1 


7 


e 


9 


K) 


II 


12 


18 


-ii 


16 


19 


14 


13 


10 


80 


^ 


a 


C3 


84 


30 


ee 


S6 


zt 


86 


25 


31 


38 


4- 


34 


35 


36 



Township— FiGUBB flO 



^79. Plotting Observations and Measurements. 

1. The hourly temperatures from 6 a.m. to 6 p.nx. Decem- 



ber 26, 1902, were: 










6 a.m. 7 a.m. 


8 a.m. 


9 a.m. 


10 a.m. 


11 a.m 


ISO r/o 


10° 


10° 


12° 


13° 



12 m. 1 p.m. 2 p.m. 3 p.m. 4 p.m. 5 p.m. 6 p.m. 
14° 14° 15° 16° 15° 14° 14° 

Draw 13 equally spaced vertical parallels 
one for each hour. Draw a horizontal line 
as OX across the parallels. Using ^" to repre- 
sent 1 ° ; measure off on the first vertical tlie 
distance 01 to represent 7°; on the second 
vertical the distance 7-2 to represent the 
second 7°; on the third parallel measure off 
tlie distance 10°, and so on. Mark the top of 
each measured vertical distance with a dot. 
FIGURE 61 Draw lines connecting these dots as in Fig. 61. 

This is called plotting the readings. 
This line shows the temperature change during the 12 hrs. 




MEASUREMENT 121 

This plotting is very much aided by the use of cross-lined 
paper, ruled into small squares. A horizontal side of one of the 
small squares might be used to represent 1 hr. and a yertical side 
to represent 1° of temperature. 

Note.— Pupils in arithmetic should have notebooks containing 
several pages of squared paper for such work as this. 

2. Read the out-door temperatures from hour to hour at your 
schoolhouse and plot your readings as above. 

3. The average hourly temperatures from 6 a.m. to 6 p.m., 
December 26 to January 2, were: 

6 a.m. 7 a.m. 8 a.m. 9 a.m. 10 a.m. 11 a.m. 
21.5^ 22.1° 22.2° 22.2° 22.5° 23.2° 

12 m. 1 p.m.' 2 p.m. 3 p.m. 4 p.m. 5 pm. 6 p.m. 
23.9° 24° 24.5° 25.2° 25° 24.7° 24.3° 

Plot these readings on squared paper, or by measurement, as 
above in Figure 61. Plot the tenths by estimate. 

4. Does the line for the averages for a week agree in a general 
way with the line for December 26? What does a comparison of 
the two lines show? 

This table gives the heights in feet and weights in pounds of 
boys and girls of ages given in the first column : 





HEIGHT VH FEKT 


Weight in Pounds 


Agb 


BOTS GIBL3 


BOYS 


GIBL8 





1.6 1.6 


7.1 


6.4 


2 


2.6 2.6 


25.0 


23.5 


4 


3.0 3.0 


31.4 


28.7 


6 


3.4 3.4 


38.8 


35.3 


9 


4.0 3.9 


50.0 


47.1 


11 


4.4 4.3 


59.8 


56.6 


13 


4.7 4.6 


75.8 


72.7 


15 


6.1 4.9 


96.4 


89.0 


17 


5,4 5.1 


116.6 


104.4 


18 


5.4 5.1 


127.6 


112.6 


20 


5.5 5.2 


132.5 


115.3 



5. Using squared paper, or making a drawing such as is indi- 
cated in Fig. 62, plot the numbers in the first column horizontally 
and^ those in the second column vertically. Draw freehand a 



122 



RATIONAL GRAMMAH SCHOOL ARITHMETIC 



lAslesJ 



FIGUBE 62 



smooth line through all the plotted points and obtain the curve for 
growth in stature of boys. At what age do boys cease growing 
rapidly in height? 

6. Using the numbers of columns 1 and 3 obtain a similar 

curve for girls. At what age do girls cease 

growing rapidly in stature? 

7. Compare the two curves and note 
when boys and when girls grow fastest? 

8. Using the numbers of columns 1 and 
4 draw the curve for growth of boys in 
weight. 

9. Use numbers of columns 1 and 5 
similarly for girls. 

10. Compare the curves of problems 8 and 9 and note any 
likenesses or differences in the two curves. What do the curves 
show about the growth of boys and girls? 

11. Twelve different rectangles each 12 in. long have, the widths 
given in the first line and the areas in the second line. 

Widths in inches.. 1" 2" 3" 4" 5" G" T' 8" 9" 10" 11" 12" 
Areas in sq. in.... 12 24 36 48 GO 72 84 96 108 120 132 144 

Two lines, OX and OY, are drawn at right 
ang:les. 12 equally spaced lines are drawn par- 
allel to OY. The horizontal distances Ol, 02, 
03 and so on denote the heights, 1", 2", 3' and 
so on. On a scale of -J" (or the vertical side 
of a small square) to 12 sq. in. of area, mark 
off the lengths la, 2b, 3c, and so on, to denote 
the numbers in the second line above. This 
gives the points a, b, c, and so on. 

Make the construction to the scale indi- figube 63 

cated or to some other convenient scale, and 
place the straight edge of a ruler along the points, such as a, b, 
c, and so on. On what kind of line do the points seem to lie? 

12. The bases of 12 triangles are each 16 in. long and the alti- 
tudes (heights) and the areas in square inches are: 



■ii, 



rnjjf t^i 'UjIfqW'^ M- ^ 



■ B » t M rt 



1" 


2" 


3" 


4" 


5" 


6" 


7" 


8" 


9" 


10" 


11" 


12 


8 


16 


24 


32 


40 


48 


56 


64 


72 


80 


88 


96 



MEASUREMENT 



123 



Using any convenient scale, plot these numbers. On what kind 
of line do all the points lie? 

13. The lengths of the sides and the areas in square inches of 
12 squares are : 

Sides.. 1" 2" 3" 4" 5" G" 7" 8" 9" 10" 11" 12" 
Areas., 1 4 9 16 25 3G 49 64 81 100 121 144 

Using any convenient horizontal and vertical scales plot these 
observations. Do the plotted points lie on a straight line? 

14. Plot these data and draw a smooth free-hand curve 
through the plotted points : 



Date of 


Population or 


Date of 


Population op 


Cknsus 


tr.S.INHlLLIOSS 


Census 


U. S. IN MXIiLJONS 


1790 


3.9 


1850 


23.2 


1800 


5.3 


1860 


31.4 


1810 


7.2 


1«70 


38.6 


1820 


9.6 


1880 


50.2 


1830 


12.9 


1890 


62.6 


1840 


17.1 


1900 


76.3 



Plot dates on horizontal, and populations on vertical, lines. 

Draw a smooth free-hand curve through the plotted points. 
By continuing the curve can you predict (tell) about what the 
population will be in 1910? 

Is the line through the plotted points a straight line? 



§80. Meaturmg By Hundredths, percentage 

1. What is j^^ of 500 mi.? 4^ of 500 mi.? ^f^ of 500 miles? 

2. What is 4^ of 200 A.? ^^ of 200 A.? ^Vd of 200 acres? 
Note.— First find jJo. 

3. What is -^VV of 100 ft.? ■^%\ of 100 ft.? ^|j of 100 feet? 

4. i of anything equals how many hundredths of it? 

5. How many hundredths of anything are the following frac- 
tional parts of it? 

i'y iy i'i f; 1 5 l> f; i\j A> A; iV; «V5 ?V> zTfy /?• 



124 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

Definition. — ^One hundredth is often written 1% and read 1 per cent- 
The sign (% ) is a short way of writing "one one-hundredth." Per cent is 
a short way of speaking **one one-hundredth." 

6. Bead and give the meaning of the following : 

2%; 6%; 8%; 12%; 12^%; 25%; 33*%; 87*%; 100%. 

7. Give the simplest fractional equivalents of (fractions that 
are equal to) these per cents : 

8. How many lb. are 50%; of 8 lb.? of 18 lb.? of 24 pounds? 

9. How many square feet are 25% of 16 sq. ft.? of 48 sq. ft.? 
of 88 sq. ft.? of 400 square feet? 

10. Heating an iron rod 100 in. long increased its length 2%. 
How many inches was the length increased? 

11. One boy threw a stone 100 ft., and another threw it 12% 
farther. How many feet farther did the second boy throw the 
stone? 

12. Referring to Fig. 1^ (page 31) point out the following 
per cents of it: 20% ; 40% ; 60% ; 80% ; 2% ; 4% ; 10% ; 100%. 

13. Draw a square and divide it by a line so as to show 50% 
of it; 25% of it; 75% of it; 12*% of it. 

14. Similarly, show the same per cents using a circle; a 
rectangle. 

15. Draw a line of any length and show 33*% of it; 66* % of 
it; 100% of it; 12*% of it; 10% of it; 20% of it. 

16. Draw a square and divide it into small rec- 
tangles as shown in the figure. Point out the following 
per cents of it: 

FIGURE 64 25%; 33*%; 50%; 66*%; 75%; 12*%; 100%. 

17. A pair of shoes costing $2 was sold at a gain of 50%. 
What was the amount gained? 

18. A boy was 50 in. in height 2 years ago. Since then he 
has grown 4% higher. How many inches taller is he now? How 
tall is he now? 

19. The height of a 14 year old boy is 60 in., and a girl of the 
same age is 2% taller. How tall is the girl? 



i^ 
















y 









MEAStJREMEi^t 1S5 

20. An umbrella was marked $1.50, and was sold at a reduc- 
tion of 20%. At what price did it sell? 

21. Eighty per cent of the cost of a girl's suit is $4.00. What 
is 1% of the cost? What is the whole cost, or 100% of the cost? 

22. An article sold for $4.00 after having been marked down 
20%. What was the price before it had been marked down? 

23. A lot was sold for $800, which was 60% more than it cost 
the man who sold it. Kow much did it cost him? 

Note.— The 60% means 60% of what it cost the seller. This cost is 
how many per cent of itself? $800 is, then, how many per cent of the 
original cost. 1% of the original cost equals what? 100% of this cost 
equals what? 

24. A thermometer reading was 120°, which was 20% higher 

than the reading taken 10 min. before. What was the previous 

reading? 

Note.— In such questions as this answer first the question, "20% of 
what?" In this case 20% of the previous reading. 

25. A man sold his farm for $12,000, which was 25% more 
than he paid for it. What did he pay for it? 

26. This year a flat rented for $55 a month, which is 10% 
more than it rented for last year. For what did it rent last year? 

27. Out of 80 games played by a championship team, 20 were 
lost. What per cent of the games were lost? What per cent were 
won? 

28. The chest measure of a boy at the close of the school year 
was 27^ in., which was 10% greater than at the beginning of the 
year. What was the boy's chest measure at the beginning of the 
year? 

29. A man paid $18 for the use of $300 for a year. What per 
cent was the sum he paid of the sum he borrowed? 

§81. Simple Interest. 

An extensive use of the system of measurement by hundredths 
is made with problems in Interest. 

Definition.— Jwferesf is money to be paid for the use of money. Two 
per cent interest means that 2^ is to be paid every year for the use of 100^ 
or $1 : $2 for the use of $100, and so on at this rate. What then would 
6% interest mean? 4% interest? 10 fc interest? 



126 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

The amount of money to be paid each year for the whole sum bor- 
rowed is called the Interest for 1 year. 

1. What is the interest on $200 at 4% for 1 year? 

2. What is the interest at 6% on $1 for 1 year? 

Note. — In computing interest a year means 12 months of 30 days each. 

3. What is the interest at 6% on $1 for ^ yr., or G mo.? for 
2 mo.? for 1 month? 

4. What is the interest at 6% on $1 for 3 mo.? for 5 mo.? 
for 7 mo.? for 14 mo.? for 16 mo.? for 34 mo.? for any number 
of months? 

5. From the answer to problem 2, find the interest at 6% for 
1 yr. on $8; on $25; on $48; on $85; on $124; on $450. 

6. If you know the interest at 6% for 1 yr. on $1, how can 
you find the interest at 6% for 1 yr. on any number of dollars? 

7. From the last answer to problem 5, find the interest at 6% 
on $450 for 2 yr. ; for 3 yr. ; for 4^ yr. ; for 12^ yr. ; for any 
number of years. 

8. Using the third answer to problem 4, find the interest at 6 % 
for 7 mo. on $48; on $450; on $75. 

9. Tell how to find the interest at 6% on any number of dol- 
lars for any number of months. 

10. The interest on a certain sum of money for 1 yr. at 6% is 
$24. What is the interest on the same sum for the same time at 
12%? at 18%? at 30%? at 60%? at 1%? at 3%? at 2%? 

11. f of the interest on a sum of money for a given time atQ^o 

equals the interest on the same sum for the same time at what 

rate per cent? 

Note.— J of a number may be easily found by subtracting from it I of 
itself. How may i of a number be found similarly? 

12. Knowing the interest on any sum of money at 6%, how 
can you quickly find the interest on the same sum for the same 
time at 7%? at 8%? at 11%? at 15%? at any rate per cent? 

13. Find the interest on $1200 at 6% for 2^ yr. ; for 2i years. 

14. A man has $350 in a bank, which pays 3% interest. To 
how much interest is the man entitled if his money has been in 
the bank 2 yr. and 4 mo.? 6 yr. and 9 mo.? 8 yr. and 2 months? 



COMMON USES OF NUMBERS 

§82. Pressure of Air. oral work 

1. When a glass is filled level with water, covered with a piece 
of writing paper, and carefully inverted, why does not the water 
fall out when the glass is held month downward? 

2. When a soft leather sucker is moistened and spread out on 
a smooth, flat surface, why does it cling to the surface even when 
the sucker is raised by being lifted at its middle? 

3. When one end of a glass tube is placed in water and you 
draw with the mouth at the other, why does the water rise in the 
tube? 

4. Fill a bottle with water, leave the cork out, and invert the 
bottle in a vessel of water. Why does the water not run out? 

6. When a glass tube with one end closed is partly filled with 
mercury and the open end dipped under the surface of mercury 
in a cup, why does not all the mercury run out of the tube? 

Note. — In connection with 5, examine a mercury barometer. 

6. When a sheet of thin rubber, or paper, is held over the 
mouth of a funnel, and the air is sucked out of the funnel 
through its neck, why does the rubber curve inward? Will it do 
this in all positions of the funnel? Why? 

These experiments show that the air presses downward, 
upward, and in all directions upon surfaces. Careful measure- 
ments have shown that the pressure of the air on every square 
inch of surface is about 15 pounds. 

Note. —This pressure is equal on all sides of surfaces, upper sides and 
lower sides, outside and inside, toward the right and toward the left. 

WRITTEN WORK 

1. Measure the length and the width of the cover of your book 
and find the downward pressure in pounds on the upper surface of 
your book when it lies flat upon the desk. Why is the book so 
easily moved about under this pressure? 

127 



128 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

2. Measure and compute the downward pressure of the air on 
the top of your desk. Can you lift this number of pounds? 
Why can you lift the desk? 

3. A room is 10' x 20' x 25'. Find the pressure of the air, in 
pounds, on the floor, on the ceiling, and on each of the four walls 
of the room. Why does not this pressure tear the walls apart? 

4. The average surface of the human body equals 20.6 sq. ft. 
What is the total pressure of the air on the outside surface? 
Why does not this pressure crush the body? 

5. Measure the length and the width of the door of your room, 
and find the air pressure on one side of the door. Why can you 
open and close the door so easily? 

6. Measure and find the air pressure in pounds on the outside 
surface of a pane of window glass. Why does not this pressure 
break the glass? 

§83. Passenger and Freight Trains. 

1. A fast train runs from Chicago to a station 356.4 mi. dis- 
tant in exactly 9 hr. What is the average rate (miles per hour) 
of the train? 

2. The train left Chicago at 6: 10 a.m. At what time did it 
arrive at the station? 

3. Another train runs 10^ hr. at an average rate of 36 mi. per 
hour, including stops; how far does it run? 

4. If the train (problem 3) started at 2: 30 a.m., at what time 
did it reach the end of the run? 

5. A traveler went by rail from Chicago to Los Angeles, 
California, in 4 da. 19 hr., at the average rate of 37.4 mi. an hour, 
including stops. How far is it from Chicago to Los Angeles by 
this route? 

6. A train ran 361 mi. in 9^ hr. What was its average rate? 

7. A freight train is running 21 mi. an hour while a brakeman 
on top of the cars walks toward the engine at the rate of 2^ mi. an 
hour. How fast does the brakeman actually move forward? how 
fast, if he walks from front to rear at the rate of 2^ mi. an hour? 

8. A conductor walks from the front to the rear of a train at 
the rate of 3^ mi. an hour while the train is running 38 mi. an hour. 



COMMON USES OF KtJMBEllS 129 

At what rate per hour does the conductor actually move forward? 
At what rate does he move in the direction of the running train 
if he walks from the rear to the front at the rate of 3^ mi. an 
hour? 

9. How may the conductor (problem 8) suddenly change his 
rate of motion from 34^ mi. to 41^ mi. an hour? How many 
miles an hour does this change of rate amount to? 

10. A passenger train running 32 mi. an hour meets a freight 
train running in the opposite direction on a parallel track 18^ mi. 
an hour. At what rate do the trains approach each other? 

11. At what rate does the passenger train (problem 10) pass 
the freight if both are running in the same direction on parallel 
tracks? 

12. A boy tries to overtake a street car by running up behind 
it. If the boy runs 10 ft. a second and the car runs 6 ft. a second, 
how soon will the boy overtake the car if he is now 20 ft. behind 
it? 

13. Two friends, coming from opposite directions, have arranged 
to meet at a certain railway station. Their trains are running at 
35 mi. and 25 mi. an hour. Not allowing for time lost in stops, how 
soon win their trains be together at the station, if they are now 
30 mi. apart and both trains reach the station at the same time? 

14. A passenger train is made up of a postal and baggage car. 
an express car, 3 common coaches, 2 chair cars, a dining car, and 
2 sleepers. The average length of a car is 61|', and the length of 
the engine and tender together is 65'. How long is the train? 

15. The weight of the engine (problem 14) is 142,780 lb.; 
of the tender, 43,200 lb.; and the average weight of a car is 
83,480 lb. What is the total weight of the train, in tons? 

16. To draw a train on straight, level track at a speed of 
40 mi. an hour, requires a horizontal pull of t^^-^ of the weight 
of the train. What force in tons would be needed to draw the 
train of problem 14 at a speed of 40 mi. an hour? 

17. An engine, 62' long and weighing 240,000 lb., draws a 
tender, 22' long, weighing 64,500 lb., and a train of 72 empty 
freight cars, averaging 36^' in length and 32,700 lb. in weight. 
How long is the train? how heavy? 



130 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

18. If it requires y/^i^ of the weight of a train to draw it os 
straight, level track at a speed of 20 mi. an hour, how many 
poands of force must the engine of problem 17 exert to draw the 
train, on such track, 20 mi. an hour? How many tons of force? 

Note.— First find jAo of the weight of the train including engine 
and tender. 

19. How much force would be needed to draw 38 cars, each 
weighing 16^ T., and each loaded with 18^ T. of coal, on straight, 
level track at a speed of 20 mi. an hour? (See problem 18.) 

20. When a train is moving 5 mi. an hour it takes a horizontal 
force of about ii^j^ of the weight of the train to draw it along 
straight, level track. On such track what force must an engine 
exert to draw a train of 68 empty freight cars, each weighing 
34,300 lb., at a speed of 5 mi. an hour? 

21. Under the same conditions as in problem 20, what force 
would be needed to draw a train of 38 cars, each weighing 34,000 
lb. and carrying a load of 58,600 lb. of coal in addition to its own 
weight? 

22. A railroad company purchased 6 locomotive engines, 
weighing, in pounds, 142,780, 142,630, 158,670, 139,790, 146,890, 
and 138,960. The tenders weighed, in pounds, 43,750, 44,200, 
45,280, 42,920, 43,650, and 44,280. The cost per pound of the 
combined weight was 13f <^. What was the total cost? 

23. The greatest horizontal pulling force that an engine can 
exert on dry, unsanded steel track in starting a train is about 
f of the part of the engine that is carried by the driving wheels. 
What is the greatest horizontal pull the engine of problem 17 can 
exert in starting a train on such track, supposing the entire weight 
of the engine is carried by the driving wheels? 

24. Answer a similar question for the engine of problem 15, 
supposing 105,750 lb. of the weight of the engine is carried by the 
driving wheels. 

§84. Train Despatcher's Report. 

A copy of a train despatcher's sheet, for a test run, showing the 
distance between stations, the schedule time and the exact mn- 
ning time of the train, the number of the cars in the train, the 
number of passengers carried each trip for 7 week days. 



COMMON USES OF NUMBERS 



131 



Pbbfobmancb op Tbaik No. 25, from July 1st to 8th, 1898. 



Sta- 
tion 



1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 



DiB- 
TANCS 

Ml 

.. 

.. 3.1 

.. 5.5 

.. 7.9 

.. 12.0 

.. 17.0 

.. 19.9 

.. 24.5 

.. 27.6 

.. 33.8 

.. 38.7 

.. 43.5 

.. 50.5 

.. 53.8 

.. 55.5 



SOBSDXTLB 
TIMB 

3:50 
3:56 
3:57 
3:59 
4:02 
4:06 
4:09 
4:12 
4:15 
4:20 
4:24 
4:28 
4:33 
4:38 
4:40 



IBT 

3:50 

3:55 

3:57 

3:59 

4:02 

4:06 

4:08 

4:lli 

4:13i 

4:18 

4:22 

4:25i 

4:301 

4.32^ 

4:35 J 



2d 

3:49 J 

3:55^ 

3:58 

4:00 

4:03} 

4:07 J 

4:10 

4:14 

4:16| 

4:21 J 

4:25 

4:28} 

4:34) 

4:371 

4:40 



4th 

8:50] 

3:56 

3:581 

4:00} 

4:03} 

4:07} 

4:09} 

4:13 

4:15 

4:20 

4:24 

4:27} 

4:32} 

4:35 

4:37} 



5th 6th 7th 



3:50 

3:55 

3:57i 

3:59} 

4:03} 

4:06} 

4:08} 

4:12 

4:14 

4:18} 

4:22 

4:26 

4:31 

4:34 

4:36 



3:50} 

3:55 

3:585 

4:00 J 

4:04 

4:08i 

4:11 

4:14} 

4:17} 

4:22 

4:26} 

4:30 

4:35} 

4:38 

4:40} 



3:50} 

3:561 

3:57} 

4:00 

4:03 

4:07} 

4:10 

4:14 

4:16 

4:20} 

4:24} 

4:28 

4:33 

4:86 

4:37} 



8th 

3:51} 

3:56 

3:58} 

4:00 

4:03 

4:07} 

4:10 

4:13} 

4:15 

4:20} 

4:23 

4:27 

4:32 

4:35} 

4:37 



Number of cars 

Passengers carried 

Running time 

Mi. per hour (average) . . 



5 
201 



7 
441 



6 
11 



5 
79 



5 
107 





113 



6 

118 



PROBLEMS 

1. By **running time" is meant the difference between the 
time of leaving* station 1 and the time of arriving at station 15. 
On a separate sheet write the * 'running time" for each day. 

2. Find the difference between **schedule time" and the time 
the train actually reached station 6 on the 2d, 4th, 6th, 7th, 8th. 

3. The train is **on time'' if it reaches a station just at 
"schedule time." Find how much th^ train was ahead of or 
behind time in reaching station 12 on each of the 7 da. Mark 
"ahead of time" results with an "A," thus: A. 2J min., and 
"behind time" results with a "B," thus: B. IJ minutes. 

4. Make and answer similar questions for other stations. 

5. How far is it from station 1 to station 15? from 2 to 9? 4 to 
11? 3 to 15? Make and answer similar questions for other stations. 

6. What is the difference in schedule times between stations 1 
and 15? 2 and 9? 4 and 11? 3 and 15? any other two stations? 

• Stops are so short that leaving and arriving times are regarded as the same. 



in 



HATIOKAL OllAMMAR SCHOOL ARITHMETIC 



7. From your answers to problems 5 and 6 find the average* 
rate of running (in miles per minute) between stations 1 and 
15, by a train running exactly on schedule time ; between stations 
2 and 9; 4 and 11 ; 3 and 15. 

8. Find the average rates of running between the same stations 
(as in problem 7) on the 6th. 

9. Find the average rates of running between stations 1 and 15 
on each day, that is, the "mi. per hour." 

10. What is the total number of cars drawn during the whole 
period? 

11. Find the total number of passengers carried. 

12. Make and solve other problems on the table. 

§85. Areas of Common Forms, oral work 

1. If one side of an inch square represents 80 rd., how many 
square rods will the inch square represent? how many acres? 

2. In a drawing of a rectangular farm to a scale 1 in. = 80 rd., 
how could you find the number of acres in the farm? 

3. If a square field containing 40 A. is cut across diagonally 
by a- railroad, how many acres does each of the triangular parts 
contain, supposing the railroad itself covers 3 acres? 

wmw" ~ 





m 

i 

1 

mi'' 



\^' 



M 



mm 



'm 






'■/,', ''/ly.'i.~hWX 





FIGUBB 65 

4. Give the areas of the cross-lined portions of the 9 forms of 
Fig. G5. 

♦ "Average rate" here means distance run divided by time of running. 



COMMON USES OF NUMBERS 



133 



PROBLEMS 

Each inoh in Fig. 66 represents 80 rods. 

1. How many 
acres are in the 
tract ABODES 
What is the farm 
worth at $95 per 
acre? 

2. The field is 
entirely enclosed, 
except along the 
river, by a barb- 
wire fence having 
4 lines of wire. 
Barb wire weighs 
about 1 lb. per rod. 
Find the cost of the 
wire at $3.50 per 
bale of 100 pounds. 

3. If posts cost 
35^ apiece and are 
set 2 rd. apart, find 
the cost of the 
posts. 

FlOURB 07 

4. In Fig. 07 1 in. represents 80 rd., or the scale is 1" to 80 
rd. How many acres are in the field ABF? What is the field 
worth at $75 per acre? 

5. Find how many acres there are in the field AFC and what 
is its value at $87^ per acre. 

6. What part of the rectangle ADBF is the triangle ABF? 
What part of the rectangle EAFC is the triangle AFC? 

7. What is the value of the field ABF at $80 per acre? 

8. What part of the rectangular tract DBCE is the triangular 
field ABC? 

9. How many rods of fence would be needed to enclose the 
field ABF? AFC? the field DBCE? 




134 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



60rd. 

10. A railroad runs straight 

across a rectangular field, dividing it 
into two triangles as shown in Fig. 68. 
How much is; each triangular part 
worth at $120 per acre, supposing 
the road takee of! equally from each 
part a strip containing 2^ acres? 

11. The railroad company fences 
both sides of its strip with 4-wire 
fences, each 96 rd. long. Posts are 
set a rod apart, and cost 25^ each. 
The wire, used costs $3.25 per bale of 

5ccl1b r= 40rd. 1^0 rd., and 2 men at $1.50 per day 

Figure 68 work 3 days to build the fence. What 

did it cost the company to fence its road? . 

12. If 1 sq. in. of the rectangle of Fig. 69 

represents f of an acre, how many acres are 
represented by the whole rectangle? 

13. If one sq. ft. of a sheet of zinc weighs 2^ 
oz., how much will pieces of the same sheet of 
the sizes shown in Fig. 'J'O weigh? 

14. If 15 sq.ft. 




,12" 
Figure 69 




te- - 16" >l 



of sheet copper 
weigh 2.5 lb., how 
much will 25 sq.ft. 
of the same sheet 
weigh? How many 
sq. ft. in 2.51b.? 



K — 36" — >♦ 

Figure 70 

15. If 5 lb. of rope cost 75^, how much would 37^ lb. of rope 
cost at the same rate? 47 lb? 

16. If the pendulum of a clock swings 40 times in 15 seconds, 
how many times will it swing in 1 minute? in 17 minutes? in 
1 hour? 

17. Two square feet of a 1-in. board weigh 9 lb. ; what would 
9 sq. ft. of the same board weigh? 72 square inches? 16 square 
inches? 



INTRODUCTION TO FRACTIONS 



135 



INTEODUGTION TO EATIO AND PEOPOETION 

Ratio 

Definition. — The ratio of one number to another is the quotient of 
the first number divided by the second. 

1. What is the ratio of A to 0? of to A? of B to A? of A 
to B? of 5 to 0? of to B? of Cto B? otCioA? of Cto 0? 
of to C? otDto C? otD to 5? of D to 0? of to 2>? 




.&'■■■ 


pr ■ 


&i 












FIGUBE 71 

2. In the second square, compare each of the parts Ey Fy and 

with ; compare with each of these parts. 

3. Compare each division with each of the others, separately. 

4. In the third square, \ oi H \% how many times J? J of / is 
what part ot H? ^ of is how many times I? 

5. What is the ratio of 3" to 1"? of 1" to 3"? of 1" to f7 of 
t" to 1"? 

6. What is the ratio of 6 to 6? of 50 to 50? of J to j? of 
a to a? of a: to a;? 

7. What is the ratio of any two equal numbers? 

^ A ^ i is a short way of writing the following 

' • t^^ expressions : 

FiouBB 73 The ratio of 1 to 4, 

1-4, 
1:4, 

1 : 4 is read ''1 to 4," and i : 1 is read "i to 1.'' 

8. What is the ratio of 1 ft. to 1 yd.? of 1 hr. to 1 da.? of 

1 nickel to 1 dime? of 1 in. to 12 in.? of 1 ft. to 3 ft.? of 12 in. 
to 1 in.? of 1 mi. to 1 mile? 









s 



























13G RATIONAL ORAMMAH SCHOOL AHITHMETIO 

The ratio of a 12-in. line to a 1-in. line is called, also, the 
measure of the 12-in. line by the 1-in. line. 

9. What is the ratio of 1 sq. ft. to a 3-in. square 
(see Fig. 73)? of 1 sq. ft. to 1 square inch? 

The ratio of a square foot to a 3-in. square is 
the measure of a sq. ft. in terms of the 3-in, square, 
that is, how many 3-in. squares in a square foot. 
Finding the ratio of the foot to the inch is the same as meas- 
uring the foot by the inch. Furthermore, to find the ratio of any 
number, or quantity, to any other number, or quantity, is to find 
the quotient of the first number, or quantity, divided by the 
second. 

The result of measuring one number by another is called the 
numerical measure of the first by the second. 

The ratio of one number to another, or the measure of one 
number by another, can be found only when both numbers are 
expressed in the same unit. 

10. Measure the avoirdupois pound by the ounce; the foot by 
the inch; the inch by the foot; the ounce by the pound; the gal- 
lon by the quart ; the bushel by the quart. 

11. Measure the rod by the yard; the mile by the rod; the 
square yard by the square foot ; the quart by the gallon. 

12. Measure 8 ft. by 4 ft. ; IG ft. by 64 ft. ; a square mile by 
a square rod; an acre by a square rod; a square mile by an 
acre. 

13. Measure 80 by 8; 80 by 4; 3 by 4; 4 by 3; 9 by 18; 18 by 
9; 125 by 25. 

14. Measure a by J; a by 2a; b by a; 2a by a; 6x by 2x; 2x 
by 6a:. 

15. If 5 apples cost 4^, what will 35 apples cost? 

Solution. — How many fives of apples in 33 apples? 

How much does one live of apples cost? 

How much t)ien do 3o apples, or 7 fives of apples, costt 
This analysis may be thought of in the form of ratios. 

Cost of 35 apples x i_ . ^ 35 a5 .^. 

Fi — I — J— i — 1 — = T^f or more briefly -z = Tk 0) 

Cost of 5 apples 4f •'5 4^ ^ ' 

and we have to find a number which has such a ratio to 4^ as 85 apples 

has to 5 apples. 

This leads us to an equation of ratios. 

Definition.-— An equation of ratios is called sl proportion. 



INTRODUCTION TO FRACTIONS 137 

§86. Proportion. 

In all these problems use the equation form of statement 
like (1) in the solution of problem 15 of the preceding section. 

1. If 1 doz.. oranges cost $0.30, what will 4 oranges cost at the 
same rate? 

2. If a yard of cloth costs $1.50, at the same rate what will ( yd. 
cost? 

3. If a 3-in. square of tin costs i^, what will a square foot qpst 
at the same rate? 

4. If 6 qt. of oil cost 15^, what will 3 gal. cost at the same 
rate? (4qt. = 1 gal.) 

5. Two rectangular flower beds have the same shape. One is 
3 ft. wide and 4 ft. long; the other is 6 ft. wide. How long is it? 

6. Two books are of the same shape but of different siaes. 
One is 5" wide x 7^" long. The other is 15" long. How wide 
is it? 

7. In two N N 
triangles of the a^ A "l? X ^/X^ 
same shape, like a/ \ b a \b " "* 
those of Fig. 74, / \ \ 

(1), i{a=4",^« ^ ^ ^ 

12", and J = 4", "^ ^^^ 

how long is B? 

8. In triangles of the same shape, like those of Fig. 74, (2), if 
a = 4", A =. 12", and * - 5", how long is B? If a = 6", b = 8", 
and B = 32", how long is A? If J = 7", B = 35", and A = 30", 
how long is a? 

9. In triangles of the same shape, as those of Fig. 74, (3), if 
A = 21", b = 9", and B - 27", how long is a? If a = 5^", A = 22", 
and B - 30", how long is b? 

10. In triangles of the same shape, like those in Fig. 74, (4), 
if a = 3", A = 9", and £7= 6", how long is c? It A ^ 24", c = 4", 
and C » 12", how long is a? 

11. Denoting the sides of two triangles having the same shape 
by letters as in problem 10 ; if a = 8", b = 12", c = 17" and B ^ 30," 
how long are A and C? 




138 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



INTRODUCTION TO COMMON FRACTIONS 

§87. FractionB as Eatios and as Equal Parts. 

I. Into how many equal parts is A (Fig. 75) divided? What is 
one of the parts called? 

m m m m ^ 

F O H I J 

FlOVBB 75 

3. One part of A equals how many of the smallest parts of B^ 
of C? of D? of E? otG? of /? of /? 

4. One part of F equals how many of the smallest parts of G? 
of H? of I? of /? of Z>? of B? 

5. Write the fraction that names one of the equal parts of -4, 
as divided in the illustration. 

6. Measure A by one of its equal parts. Ans, 2 halves or f . 

7. Measure the shaded part of B by one of its 4 equal parts. 

Ans, f, read "2 fourths." 

8. Measure the unshaded part of B by the same unit. 

9. Using one of the 8 equal parts of (7 as a unit, express the 
shaded part of in this unit; the right half of the shaded part. 

10. Express E in terms of its smallest unit. 

II. Express each figure in terms of its smallest unit, as follows : 

(1) The whole square, F, G, H, /, /. 

(2) The shaded part of " " " '' " 

(3) The unshaded part of '* " " '* '' 

t^\ <( <c ^. <c cc . cc c< 

(4) The shaded part of G in the smallest part of 7; 

(5) The unshaded part of / and of G in thesmallest part of/. 

Definition.—} means 3 of the J's (one fourths) of some number, or 
•^ured quantity. The J is called the fractional unit. The fractional 
of any fraction is one of the equal parts expressed by the fraction. 



INTRODUCTION TO FRACTIONS 139 

12. What is the fractional unit of each of the following frac- 
tions: f, I, f, 1, 4, h h h ii, h h h h h 1%. A. A, t'V? 

13. What is the fractional unit of a fraction having the num- 
ber 7 below the line? the number 6? 9? 12? 15? 25? 18? 11? 64? 

14. How many fractional units are expressed by the 1st frac- 
tion of question 12? by the 2d fraction? the 3d? 4th? 5th? 6th? 
7th? Draw a square and divide it free-hand by lines showing 
the meaning of each of the first 8 fractions of problem 12. 

15. What does the number written below the line of a fraction 
express? 

16. What does the number written above the line of a fraction 
express? 

Definitions.— The number above the line is called the numercUor 
(meaning nitmbcrer). 

The number below the* line is called the denominator (meaning 
namer). 

17. What are the fractional units of fractions having the fol- 
lowing denominators : 

- 3? 7? 12? 18? 24? 125? 75? 19? a? x? 

18. Point out the fractions of question 12 that have the same 

fractional units. 

Definition. — ^The numerator and the denominator are together called 
the terms of a fraction. 

19. If each half of a square is divided into 2 equal parts, into 
how many equal parts is the whole square divided? 

20. Into how many equal parts is a whole divided, if each half 
of it is divided into 3 equal parts? into 4 equal parts? into 5? 6? 
7? 8? 9? 10? 11? 25? 

21. Into how many equal parts is a whole divided, if each third 
of it is divided into 2 equal parts? into 3 equal parts? into 4? 5? 
6? 7? 8? 9? 10? 12? 15? 20? 30? 

22. Into how many equal parts is a whole divided by dividing 
each of its fifths into 2 equal parts? 3 equal parts? 4? 5? 6? 7? 
8? 9? 10? 20? 30? 

Give solutions of such as the following and explain: 

23. Express as sixths, i, f , |, i^, |. 

24. Express as twelfths, i, J, |, i, J, J, #. 



140 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

25. In question 12 point out pairs of fractions that are not 
expressed in the same fractional unit, but may easily be so 
expressed. 

26. Express the following pairs of fractions as equivalent 
fractions having the same fractional unit, or, what is the same 
thing, having a common denominator: 

J and f ; i and ^ ; | and f ; | and f ; f and |; i and -f^. 



1 . 1 . 1 . 1 . 1 . t . 1 . 1 . 1 


^e 


'A 'A 




¥ 










1 . . . 1 . . . 1 . , . i 




Vl2 


^ 


'A 










1 , 1 . 1 . 1 . 1 . 1 • 1 . t . 1 . 1 



COMMON FEACTI0N8 

REDUCTION OF COMMON FRACTIONS 

§88. To Eeduce Fractions to Higher, Lower, and Lowest Terms. 

^^ 1. Express-Jas4:ths;8th8: 

16ths. 

2. Express J as 6ths ; 9ths 
12ths; 18ths. 

3. Express as 3ds f ; f 

y^ y^ 4. Express as 7ths /, ; H 

FIGURE 76 IJ. ||. 

5. Compare the values of these fractions after reducing each 
to its lowest terms (that is, to the smallest possible whole num- 
bers for numerators and denominators) . 

1 5 I J W> \\\ \\\ tVttj \%%' 

6. Express! as lOths; 15ths; 20ths; 35ths; 75ths; lOOths. 

7. By what must you multiply the numerator of f to make 
the numerator the same as that of |? By what must you multiply 
the denominator of \ to make the denominator the same as that 
of I? Draw a square and divide it free-band to show the com- 
parative sizes of I and | (see square (7, Fig. 75, §87). 

8. By what must you multiply each term of | to obtain ^^ 
W f ^? tVi7? f H? What, then, is the value of each fraction of 
problem 5? 

9. How then may the value of a fraction be expressed ia 
higher terms? 



7 



COMMOK FRACTIONS 141 

A fraction may be expressed in- higher terms ^ without chang- 
ing its value^ by multiplying both terms by the same number, 

10. Express the following fractions in higher terms: 

I; -l;l;il;il;T'j; If 

11. How can you obtain the terms of | from the terms of f ? 
of ■^%? of IJ? of ^Vo? 

12. What fraction do you obtain by dividing each term of 
j*g by 4. What, then, is the value of /^ as compared with the 
fraction ^? 

13. By rising a rectangle, divided as in Fig. 
77, show that ^^ of the rectangle equals \ of it. 

14. In what lower terms can f ^ be expressed? fiqubb 77 
T*A? iS^ Show that f J = f by means of a properly divided 
rectangle. 

15. What effect on the value of a fraction is made by dividing 
each of its terms by the same number? 

16. How may the value of a fraction be expressed in lower 
terms? 

Principle I. — Multiplying or dividing both terms of a frac- 
tion by the same number changes the form of the fraction without 
altering (or changing) its value. 

Fractions are most easily used when their numerators and 
their denominators are the smallest possible whole numbers. 

Definition. — The fractions are then said to be in their lowest terms. 

Among the many ways in which Jf can be written are these: 

17. Show how } is obtained from J| by the use of Principle I. 

18. Give the values of these fractions in their lowest terms: 

tV; I> Isj tV> II 'j ft- 

To obtain the value of a fraction in the smallest possible terms, 
it is necessary to divide both terms by the largest exact common 
divisor of both terms. 

Definition. — ^This divisor is called the greatest common divisor of the 
terms. It is indicated by the initial letters G. C. D. 



142 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

Dividing both terms of a fraction hy their 0, C, D. reduces 
the fraction to its lowest terms, 

19. Why are these fractions in their lowest terms? 

t; i; *; i; I; !; \h if; i*. 

Among these fractions is there a factor common to any numer- 
ator and its denominator? 

, Definition. — Two numbers that have no common factor, except 1 , are 
said to be prime, to each other. 

20. Reduce these fractions to their lowest terms : 

ii;ll; i-»; ll;.fl;T¥r; Hh 

§89. Factors, Prime and Composite. 

Note. — Review tests of divisibility §57, pp. 73, 74, and use them 
. through this section and the next, when searching for factors. 

1. Write down all the factors, or exact divisors, of 36 and 

of 48. 

Definition. — By factor is here meant exact divisor, or a divisor that 
is contained without a remainder. 

Convenient Form 

2 )36, 4 8 The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 

2 )18, 24 18^ and 36. 

3)9^ The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 

2. What factors are common to both 36 and 48? 

3. What is the greatest factor that is common to 36 and 48? 

4. Arrange the factors of 42 and 105 as the factors of 36 and 
48 are arranged in problem 1, and answer questions like 2 and 3 
for the factors of 42. and 105. 

5. Give the factors that are common to the 3 numbers, 18, 24, 
and 36. Give the greatest factor common to all three numbers. 

6. Answer questions like those of problem 5 for these numbers : 
(1) 12, 72, and 84; (2) 42, 98, and 168; (3) 36, 84, 96, and 108. 



COMMON FRACTIONS 143 

7. Are there factors of 5 other than itself and 1? of 7? of 13? 

Definitions. — A number that has no factors except itself and 1 is a 
prime number. A number that has factors beside, itself and 1 is a com,' 
posite number. 

8. Name the prime numbers from 1 to 100; the composite 
numbers from 1 to 100. 

Query. — Is 2 a prime or a composite number? 

Definitions. — A number that can be exactly divided by 2 is an even 
number. All numbers that can not be exactly divided by 2 are odd numbers. 

9. How can an even number be quickly recognized? (See test 
of divisibility by 2, p. 73.) 

10. Name the even numbers from to 50; the odd numbers. 

11. Tell what numbers of these are (1) even, (2) odd, (3) 
prime, (4) composite: 5, 8, 9, 2, 21, 15, 19, 26, 27, 38, 41, 42. 

12. Mention some numbers that are both odd and composite ; 
even and composite ; odd and prime. 

13. Factors of 28 are 7 and 4. As 7 is a prime number it is 
called a prime factor. What is a prime factor? 

14. Write the prime factors of 21; of 24; of 25; of 27; of 30. 

Note. — Write out the prime factors of these numbers as they are 
here written out for 96: 

96 = 2x2x2x2x2x3. 

15. How many times does 2 occur as a prime factor in 96? 
This may be indicated by writing 96 thus: 2^x 3. The small 6 
written to the right and above the 2 is to show how many times 2 
is to be used as a factor. 

§90. Greatest Common Divisor by Prime Factors. 

1. What are the prime factors of 36 and 48? (See problem 1, 
§89.) Write 36 in the form given for 96 in problem 15 of the 
last section. Write 48 also in this form. 

2. Will each of the common prime factors of 36 and 48 divide 
the G. C. D. of 36 and 48? Will the product of all the common 
prime factors divide the G. C. D. exactly? 

3. Answer questions like 1 and 2 for the numbers 42 and 105 ; 
for 96 and 21^ ; for 75 and 250. 



144 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

4. When any common prime factor occurs oftener than once in 
one or both of the numbers, how often does it occur in the G. C. D. 
of those numbers? Answer by examining these pairs of numbers : 

(1) 12 and 48; (3) 75 and 250; 

(2) 54 and 405 ; (4) 98 and 343. 

5. Make a rule for finding the G. C. D. of two or more numbers 
from their common prime factors when no common prime factors 
are repeated in any of the numbers. Test your rule by finding 
the G. C. D. of 30 and 42; of 105 and 231 ; of 30 and 70. 

6. Make a rule that will give the G. C. D. of two numbers 
when one or more of the common prime factors is repeated in 
one or both of the numbers, and test your rule by finding the 
G. C. D. of 72 and 108; of 288 and 648; of 675 and 1125. 

7. Find the G. C. D. of 792 and 1080. 
Convenient Form 

2) 7d2 1080 

792 = 2» X 32 X 11 
1080 = 28 X 3a X 5 
G. C. D. = 2^X3=^ = 8X9 = 73 
Ti 3 )l5" 

5 
Principle II. — The G, C, D, of two or more mimbers equals the 
product of all the prime factors common to all the numbers^ each 
commo7i prime factor being used the smallest number of times it 
occurs in any one of the given numbers. 

8. Find the G. C. D. of the sets of numbers in problem 6, §89. 

9. Find the G. C. D. of the following, applying tests for divisi- 
bility: 

(1) 100 and 120; (5) 108, 162, and 216; 

(2) 54 and 144; (6) m, 176, and 286; 
• (3) 225 and 315; (7) 315, 630, and 756; 

(4) 215 and 1935; (8) 162, 1134, and 1458. 

10. Reduce these fractions to their lowest terms by dividing 
both terms by their G. C. D. : 



2) 396 


540 


2)198 


270 


3)99 


135 


3)33 


45 



(1) \%; 


(4)H«; 


(7) 1!?; 


(10) ^Vt; 


(2) if; 


0) m\ 


(8)VA; 


(n).TRA; 


(3) n-, 


(6) m; 


(f) iSI; 


(12) M- 



COMMON FRACTIOKS 145 

§91. froblems. 

In each case when the answer to the problem is a fraction, or 
ratio, it rriust be expressed in its lowest terms. 

1. A man works 48 da. out of 64 da. What part of 64 da. 
does he work? 

2. A grocer bought 24 boxes of oranges and sold IG. "What 
part of his purchase was sold? 

3. Out of 56 bu. of potatoes a huckster sold 49 bu. What 
part of his potatoes was sold? 

4. A street car on one line makes a weekly average of 72 trips. 
A car on another line makes an average of 144 trips. What is 
the ratio of the former to the latter? 

5. A man pays $50 for wood and $125 for coal in one season. 
Find the ratio of the cost of the wood to the cost of the coal. 

6. Out of 1000 ft. of lumber purchased, 500 ft. were used for 
the flooring of two rooms. Let x equal the ratio of the number of 
feet of lumber used to the total number of feet purchased. Find 
the value of x. 

7. Out of 126 bu. of oats, a livery man fed 63 bu. in 1 wk. 
Let y equal the ratio of the quantity of oats bought to the quan- 
tity of oats fed. Find y. 

8. In 667 lb. of sandy loam there were 377 lb. of sand and 
gravel. What part of the soil by weight was sand and gravel? 

9. The human body needs about 94.5 oz. of water and solid 

food each day, of which 64.8 oz. should be water. The water is 

what part, by weight, of the totaV quantity of solid food and water? 

Suggestion. — Both terms of the fraction may be multiplied by 10 
without chaDging the value of the fraction. Then find the G. C. D. of 
the new terms and divide both new terms by it. 

10. The total population of a certain city is 36,720, of which 
12,243 are colored and 10,017 are foreign. What part of the entire 
population is colored? What part is foreign? 

11. What part of the total population is made up of colored 
and of foreign persons? 

The solution of problem 10 will show that J of the population 
is colored, and that -^j of it is foreign. AVe can solve problem 11 
if we can find the value of y + -^i . 



146 BATIONAL GRAMMAR SCHOOL ARITHMETIC 

12. A boy spent f of his money and gave away J of it ; what 
part of it did he have left? 

To solve this problem we must know how to add and subtract 
fractions. This is what we shall study next. 

§92. Fractions Having a Common Fractional Unit (a Common 
Denominator). 

1. Complete these equations : 



|yd. + |yd. = 


fgal. + fgal. = 


6 + 6 " 


M^+tA = 


i^ + tV = 


^ + ^ = 

X~ X 


|hr. + ihr. = 


A+A- 


Ob a+6 


4 wk. + f wk. = 


jV + iV- 


z~ u 


1 bu. + 1 bu. = 


4r+4 = 


2. + ^.= 

n ' n 


Note.— -J- is read, ' 


•5 divided bya,"and~ is 


read, "x divided by z. 



^^ is read, *'a plus 6 divided by c." 

2. Make a rule for adding fractions having a common denom- 
inator. 

3. Complete these equations: 

*-!= 1-1= T'V-f'T= tA-*A = 

tV-^= 4-1= i-|= 43da._VSrda.- 

ab mn caB__ ax 

c '~' o ~' X X ~' r r " n n " 

Note.— ^ is read, "a minus h divided by c." 

4. Make a rule for finding the difference between two fractions 
having a common denominator. 

5. Denoting any two fractions having common denominators 
by 7 and —, state in symbols : 

Principle III. — The sum^ or the difference^ of any two frac- 
tions having common denominators equals the sum^ or the difference^ 
of their numerators^ divided hy the common denominator. ' 

§93. Fractions Easily Beduced to Common Fractional Vnit. 

1. } equals how many 12ths? Sths? 16ths? ^Oths? 64thB? 
lOOths? 



COMMON FRACTIONS 147 

2. jV hr. + i hr. = a; hr. What is the value of x? 
Solution.— /, hr. + ^ hr. = j} hr. = | hr. a? = f . 

8. yV hr. - } hr. = y hr. What is y'^ 

4. Find these sums and differences : 

Fractions, whose fractional units are not the same, must be 
expressed in a common unit before they can be added or sub- 
tracted. To do this we must find a fractional unit, whose denom- 
inator can be exactly divided by each of the given denominators. 
It will be the simplest always to select the fractional unit whose 
denominator is the least number that each of the given numbers 
will divide. 

Definition. — A denominator which is common to two or more frac- 
tions is called a common denominator. When this common denominator 
is the least number that can be found whicli may be used as a common 
denominator of the fractions, it is called the least common denominator^ 
and is written L. C. D. 

Before studying more diflBcult fractions we must learn how to 
find least common denominators. We begin by a study of multi- 
ples, common multiples, and least common multiples of numbers. 

5. Five measures of last year's growth of twigs from the lower 
branches on the north side of an oak tree are: 5^", GJ", 4J", 5J" 
and 6J". What is the average growth? 

6. From the lower branches on the east and west sides of the 
same tree the measures are: 6j", 4f", (3i", GJ" and Gf. AVhat 
is the average growth of twigs on the east and west sides of the 
tree? 

7. For twigs on the south side of the same tree the measures 
are: 6 J", 6j", GJ", 8 J" and ^'\ What is the average growth of 
twigs on the south side of the tree? 

8. From the middle branches of the oak tree 5 measures of 
branches from the north side are: 5i", 4i", 7^", 3j" and 5J"; 
of branches from the east and west sides are: 4J", 7^", 5^", 7 J" 
and 6f"; of branches from the north side: 5^", Gf, 7f', 6f" 
and 7i". How much greater is the average growth of twigs on 
the south side than the average growth of those on the north side 
of the tree? than twigs from the east and west sides? 



148 NATIONAL OHAMMAR SCHOOL ARITHMETIC 

§94. Multiples. 

ORAL WORK 

42 yd. is exactly measured by 7 yd., 3 yd., and 2 yards. 

$21 is exactly measured by $7 and $3. 

56 pk. is exactly measured or divisible by 7 pk. and 8 pecks. 

42 is a multiple of 7, 3, and 2. 21 is a multiple of 7 and 3. 
56 is a multiple of 7 and 8. What are some of the multiples of 
3 and 5; of 5 and 7; of 2 and 11? 

1. What then is a multiple of a number? 

Definition. — A number that can be exactly divided by another 
number is called a multiple of the latter number, or a multiple of a num- 
ber is found by multiplying it by some whole number. 

Following is a list of multiples of 2, from 2 to 36, inclusive: 

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 3G. 
Following is a list of all the multiples of 3, to 36: 

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36. 

2. Why is each number in the first row a multiple of 2? 

3. Passing along the rows underscore the numbers that are the 
same in both rows. 

4. In what way are these multiples of 2 and 3 different from 
the rest? Ans. They occur in both rows. 

5. What is a common multiple of two numbers? 

Definition.— A number that can be exactly divided by two or more 
numbers, is called a common multiple of those numbers. 

6. Rewriting the common multiples of 2 and 3 we have: 

6, 12, 18, 24, 30, 36. 
Can yon supply a few more common mnltiples here without 
extending the rows above problem 2? 

7. What number, besides 2 and 3, will exactly divide all the 
common multiples of 2 and 3? 

8. On the blackboard write out rows of multiples of 3 and of 5, 
like those above, underscoring, or writing in colored chalk, the 
multiples that are common to both rows. 

9. Will the least of the common multiples of 3 and 5 divide 
all the other common multiples? What is the least common 
multiple of two numbers? 



COMMON FRACTIONS 149 

Definition. — The least common multiple of two or more numbers is 
the least number that is exactly divisible by each of the numbers. It 
is usually written L. C. M. for brevity- 

§95. Finding the L. C. M. 

WRITTEN WORK 

1. Make lists of the multiples of 2, 5, and 7, and find their 
L. C. M. 

2. In the" same manner, find the L. C. M. of 3, 4, and 5. 

3. Note that all the given numbers in problems 1 or 2 are 
prime to each other. In such a case how can the L. C. M. be 
found from the numbers? 

TTie L. C, M. of two or more Clumbers, all prime to each other ^ 
is their product. 

4. Illustrate the truth of this statement by two sets of num- 
bers of your own selection, one to contain two numbers, and the 
other three, the numbers of each set being prime to each other. 

5. 4, 8, 12, 16, 20, 24 are multiples of 4; and 6, 12, 18, 24, 30 
are multiples of 6. What is the least common multiple of 4 and 6? 

6. This number is the product of 4 and 6 divided by what? 
2 is the common factor of 4 and 6. 

7. Are 4 and 10 prime to each other? What is their greatest 
common factor or divisor? 

8. 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 are multiples of 4, and 
10, 20, 30, 40 are multiples of 10. What are some common mul- 
tiples of 4 and 10? What is their least common multiple? 

9. Divide the product of 4 and 10 by 2, their G. C. D. What 
is the result? Compare this result with the last answer to 
problem 8. 

The L, C. M, of two numbers not prime to each other is their 
product divided by their greatest common divisor {0. C. z).), 

10. What is the greatest common factor of 15 and 20? What 
is the product of 15 and 20? How can you find the L. C. M.? 

11. In the same manner find the L. C. M. of 8 and 10; of 12 
and 24; of 15 and 25; of 24 and 60; of 40 and 36. 



150 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

§96. Shorter Process for Three or More Nnmbert. 

ORAL WORK 

2. In a certain number there are 8 24'8; how many 12's are 
there in the number? how many 8's? how many 6's? 

2. In a certain number, x, there are 15 6's. How many 3's 
are there in a;? how many 2's? 

3. If 6 exactly divides a certain number, rr, what other num- 
bers also exactly divide xt 

4. If any composite number exactly divides a number, y, what 
other numbers also divide y? 

Any multiple of a number is divisible by all the prime factors 
of that number. 

WRITTEN WORK 

1. Find the L. C. M. of 160, 504, and 540. 

Solution.— 150 = 2x3x5x5 =2x3x5*. 

504 = 2X2X2X3X3X7 = 23 X32X7. 
540 = 2x2x3x3x3x5 = 2^ X3»X 5. 

The L. C. M. is a multiple of each of these numbers severally. 
By the above principle it must then contain the prime factors 
2, 3, 5, and 7. But, since 25 = 5* is a factor of 150 the L. C. M. 
must contain 5 twice as a factor. Since 8, or 2', is a factor of 
504, the L. C. M. must contain 2 as a factor three times. How 
many times must the L. C. M. contain 3 as a factor? How many 
times must the L. C. M. contain 7 as a factor? The number 
that contains just these factors and no others is 2x2x2x3x 
3x3x5x5x7 = 2»x3»x5«x7 = 37,800. 

37,800 is the least common multiple, because it contains the 
necessary factors and no others. Any other common multiple 
must contain all these factors and some other, and it would there- 
fore be larger than 37,800. 

What prime factors must be in the L. C. M. of three numbers? 

If a prime factor is repeated in one or more of the numbers, 
how often must it occur in the L. C. M.? 

Principle IV. — TJie L, 0, M. of two or more numbers is the 
product of all the different prime factors of all the numbers^ each 
factor occurring the greatest number of times it occurs in any one 
of the numbers. 



3) 6, 


18. 


27, 


36 


2) 3, 


6, 


9, 


12 


3) 1, 


3, 


9. 


6 



COMMON FRACTI0K8 151 

2. Find the L. 0. M. of these numbers and show your work : 

(1) 9,12,18; (4) 15,25,42,50; 

(2) 6,24,30; (5) 7,28,24,42; 

(3) 18, 30, 36; (6) 6,18,27, 36. 

3. The work can be shortened by this arrangement. Let 
us solve problem 2, (6). 

^ „ Explanation.— Select any prime 

Convenient Form number, as 8, that will divide two 

or more of the numbers whose L. C. 
M. is sought. Divide it into all the 
numbers that are exactly divisible 

by it, writing in the line below the 

1, 1, 3, 2 quotients and any numbers the chosen 

T n ur q\/On/QvQ\xo -iaq divisor does not exactly divide. 

L. C. M. -3X2X3X3X3- 108. ^^^^ another prime number and 

do as before. Continue until the numbers last brought down are all 
prime. The continued product of the divisors and the numbers remain- 
ing in the last horizontal line is the L. C. M. 

A little study will show this to be merely a more convenient way of 
finding the factors described in Prinbiple IV. 

The least common denominator defined on page 147 is the 
L. G. M. of all the denominators of the fractions. 

4. Find the least common denominator and then add 

(1) Ai A» A> and if; (3) i, -f^, /y, and ^^; 
(^) 4> ijVi A-> and ^ji • (4) |, V^^, j\, and ^V- 

§97. Definitions and Principles. 

Thus far we have had to do with two kinds of numbers: 
(1) whole numbers, or Integers^ and (2) fractional numbers, or 
j^ractions. 

Definitions. — A proper fraction is a fraction whose numerator is less 
than its denominator: as |, |. \. 

An improper fraction is a fraction whose numerator is equal to, or' 
greater than, its denominator : as |, |, I, f . 

A mixed number is a number, such as 2}, 12}, 7f , that is composed 
of an integer and a fraction. 

1. Name the proper and the improper fractions, the mixed 
and the whole numbers : 

f, f, iV, 5, V-, I, I, 1^, 3xV, 6.75, ^^, 2. 

2. Make and give an example of each class. 

3. How many times is 1 contained in 3? in 18? in 25? in any 
number? 



152 RATIONAL GRAMMAB SCHOOL ARITHMETIO 

We may write this in symbols, thus: 

1 = 3; Y=i8; V=2S; 7-«. 

Any whole number may he written in the fractional form by 
writing 1 for its denominator. 

4. Write the following numbers in fractional form: 
12; 24; 32; 68; a; x 

6. How many halves are there in 1? in 2? in 5? in 2J? in 5J? 

6. How many 5ths are there in 1? in 2? in 4? in 6? in 6|? in 
71? 

7. Change the following integers into 6ths: 2; 5; 16; 29; 120. 

8. Change the following numbers into 12ths: 2; 8; 12; 20; 
56; 128. 

9. How may any whole number be changed into 12ths? into 
9ths? into 24ths? into a fraction having any given denominator? 

Principle V. — Any integer may he expressed in the form of 
a fraction having a given denomhiator hy multiplying the integer 
by the given denominator and writing the product over that denom^ 
inator. 

10. Express 6, 18, 17, 22, 39, 28 as 4ths; as 7ths; as lOths; as 
lOOths. 

11. Change into 6ths, 2; 4^; 24|; 37f 

12. Change the following numbers into improper fractions 
having 7 for a denominator : 

3;^; 24f; 106f; 234^; 648. 

13. How can you change 'any mixed number into an improper 
fraction whose denominator is the denominator of the given frac- 
tion? 

Principle VI. — A mixed number may be expressed as an 
improper fraction by multiplying the whole number by the denom- 
inator^ adding the numerator to the product^ and writing the sum 
over the given denominator. 

14. Reduce to improper fractions: 

H\ 6i; 12i; 18i; 25f ; 328|; 609^. 



COMMON FRACTIONS 



153 



t4^ 



15. Express V- as a whole number. Change V- ^ * mixed 
nnmber. 

16. Change to whole, or mixed, numbers the following improper 
fractions: 

f; J; J; V-; V-; V; ¥; V; V; If; It; V; V; i 

17. How may any improper fraction be changed to a whole, 
or mixed, number? 

PRii;rciPLE VII. — An improper fraction may be changed to a 
whole^ or mixed^ number by making the indicated division. 

18. Reduce to whole, or mixed, numbers the following: 

I; V-; If; n^; V-; IM; HF- 

§98. Addition of Fractioiis. 

1. If the numbers written on the lines of the drawing are the 
lengths in feet of the inside walls of the house, 
how many feet of lumber would there be in a 
baseboard not over V thick, 1 ft. wide extend- "'- l^^ 

ing entirely around the inside of the house, 
deducting 6 board feet for doors? 

To solve this problem we add first all the '*^^ 12 j4 

whole numbers and then all the fractions. 
We must now learn how to add such common 
fractions as occur here. This problem will be 
solved later. (See problem 9 below.) 

2. A man owns f of an acre in one block and | of an acre 

in another block ; how much land does he own? 

This problem requires us to add f and i. 

I. Geometrical Solution.— Fig. 
79 (a) is divided by the vertical lines 
into 3 equal parts and by the hori- 
zontal lines into 8 equal parts. Into 
how many e(][ual parts do the 2 sets 
of parallel lines divide the square 
(a)? 

Show by Fig. 79 (a) that i = 15- 
Show by Fig. 79 (5) that { = ii- 



r 
9^ 



2^4 



FlOUBB 78 



(a) 



(0) 



Fioniai ra 



| + f = H + H = H = Hi- 



Ans. \\\ aor99- 



154 



EATIONAL GRAMMAR SCHOOL ARITHMETIC 



n. ARiTtiMETiCAL SOLUTION.— What is the least common denomi- 
nator of f and f ? What, then, is the largest fractional unit (therefore 
having the least denominator) in which both } and i can be exactly- 
expressed? 

i = how many 24ths? I = how many 24ths? 

164-21 _ ,,, . -,, 

Ana. m acres. 



Add 4, ij, f , and '^^. 



7 




12 



*< 



(b) (c) 


(d) 


fi = A 1 = 1! 


A = H 


FlGUBB 80 





(a) 



I. OE0METRICAL SOLUTION.— Show from Fig. 80 (a), that f = H ; from 
Fig. 80 (6), that A = A; from (c) that J = ?f ; from (d) that ft = fj. 

U + A -h S5 + fl = w = m = iii. 

II. Arithmetical Solution.— The least common denominator of the 
fractions is 84. 

3X12, 2X4, 2X28, 5 X 7 _ 36 + 8 + 56 + 85 

84 



f + A+i+i% = 



185 

* 84 



7X12^21 X4 ' 3X28 ' 12X7 



= ili=lH. 



The last step consists in reducing the result to its simplest form. 

4. ^ of the weight of a quantity of soil was gravel, and f 
was sand; what part of the soil hy weight was sand and gravel? 
Draw a figure and give the geometrical solution. Give also 
the arithmetical solution. 

5. One side of a triangle is 2f in., another is 2f in., and the 
third is 3 1 in; how long is the perimeter of (distance around) the 
triangle? Add first the whole numbers, then the fractions, and 
finally add the sums. 

G. One side of a 4-sided figure is 6f in. long, a second side 
Is 3t in., the third Tf in., and the fourth T^V in. What is the 
perimeter of the figure? 

7. A coal dealer bought 4 carloads of coal of the following 
weights: 3UJ T., 27| T., 29| T., and SOJ T. .Find the com- 
bined weight, in tons. 



COMMON FRACTIONS 155 

8. Solve these problems as rapidly as you pan work accurately, 
asing your pencil merely to write the products and the sums : 

(1) I + * =? (4) 4+iH = ? (7)H + i = ? 

(2) i + i =? (5) H+tV=? (8) -^+1- = ? 

(3) A + A = ? (6) iV + if =? (9) |- + f=? 

9. Solve problem 1 of this section. 

§99. Subtraction of Fractions, 

1. A boy had $| and spent t^; what part of a dollar did he 

have left? 

Arithmetical Solution.— The least common denominator is 2 X 6 
«10. 

l=A;l = iV Then, 1-1= ft- A = iV ^n«.$ft 

2. William is 4f ft. tall and James is 4^^ ft. tall; who is the 
taller and by how much? 

3. A man owned {i A. of land and sold f A. ; how much land 
did he then own? 

4. From f lb. of loam f lb. sand was removed; how much 
of the loam remained? 

5. A man having 14^ paid a debt of $2i; how much money 
had the man after paying the debt? 

Solution.— J = 5^, } = Jg. As ^ is less than If, write 4a^ in the 
form3|}. The problem is then 8iJ--2l8 = lA for 8 — 2=:1 and |J — 
18 = A- 

6. Solve these problems: 

(1) 13i - 7f = ? (4) 128f - 97if = ? 

(2) 284 -19f = ? (5) 639f -598jV = ? 

(3) 30^-16i = ? ' (6) 1217if -989i| = ? 

7. A tree 72 J ft. high is broken off 28y\ ft. from the top; how 
high is the stump? 

8. A 5-cent piece weighs 73} gr. and a quarter dollar weighs 
96^ gr. ; how much more does a quarter weigh than a 5-cent 
piece? 

9. The dime weighs 38j\ gr. Before 1853 it weighed 41J gr. 
By how much was the weight of the dime reduced in 1853? 



156 rational grammar school arithmetic 

10. A silver dollar weighs 412^ gr. and a double eagle 
weighs 516 gr. What is the difference between the weight of 
a double eagle and that of a silver dollar? 

11. What is the difference between the weight of the half 
dollar (192^^ gr.) and that of the quarter dollar (96^1^ gr.)? 

12. Write out rapidly the values of these sums and differences : 



(1) J+^ =? (6) 1 + f =? 
(2)i-iV=? (7)f-T\=? 
(3)+-T>j = ? (8)t\ + t\ = ? 
(*)^+V=? (9)-7+T=? 
(5)V-V=? (10)=--^=? 

13. Make a rule for finding quickly the 
of any two fractions. 

Note.— This latter rule may be shown thus, 
1st num. X 2d den. + 2d num 


(11) f + 1 = ? 

(12)t\-tV = ? 

(13) TV+l»tF-? 

(14)^+-? =? 
(15) ^-f -? 

sum, or the difference, 
X 1st den. 


and 

HiflpArAnruit — 


1st den. X 2d den. 
1st num. X 2d den. — 2d 


num. X 1st den. 



1st den. X 2d den. 
The result must always be reduced to the simplest form. 

14. In three successive runs, a train loses i, i, and i hr. 
How much time does it lose in all? 

15. f of a man's salary is spent for clothing, i for board and 
lodging, ^ for books and stationery, yV for traveling expenses. 
What part of his salary was spent for these purposes? what part 
remained? 

16. In the six days of one week, a man works 8^ hr., 9J hr., 
8| hr., 9 J hr., 8f hr., lOf hr. What is the whole number of 
hours of work for the week? What are his wages at $.30 an hour? 

17. Four children are to do a piece of work. Three of them do 
f , J, and f of it. What part is left for the fourth? 

18. I, i, and -^j of a man's money are invested in three different 
enterprises. What part of his money is invested? What part is 
free? 

19. A merchant sold y\ of a gross of buttons to one customer, 
and a of a gross less to another. What part of a gross did the 
second customer buy? 



COMMON PRACTIOKS 157 

20. A owns yV and B ^^ of an estate. How much more of the 
estate does A own than B? 

21. A tank of oil is f full. If ^ of the contents of the tank is 
drawn off and then -^ of the remainder, what part is left? 

22. A owns ff of a mill. B's interest is J f of the mill less 
than A'b. What part of the whole does B own? 

23. At $1 a day how much money does a man earn by working 
H da., ^ da., and 8| days? 

24. A boy lives where school is taught 1100 hr. a year. He is 
compelled by sickness and other causes to lose from month to 
month the following numbers of hours : 

115t; 39i; 15f ; 6H; 4f ; 3f ; 6|; 18J; 89^. 

How many hours did he lose in all? What part of the whole 
school year did he lose? 

Note. — In adding the fractions of problem 24 group together fractions 
whose denominators are the same, or are easily made t)ie same. 

§100. Multiplying a Fraction by a Whole Number. 

1. 6 times 3 mi. = how many miles. 

2. 8 X 4 yd. = ? yards. 
8. 8 X 4 fifths = ? fifths. 

4. 6x4 = —; what is the value of a;? 

5. 9 X ^T = -—■; what is the value of a:? 

6. A line is divided into 11 equal parts, and 5 of them are 
taken. What fraction represents the part of the whole which is 
taken? 

7. What fraction would represent 3 times this part? 

8. Replace the letter in each of these equations by its correct 
value; the sign (x) should be read "times" in. these problems. 

(l)6xf = f (5)5xHf (9)2xi\=^ 

(a)4xj=f (6)3x| = f (10)6x| = f 

(3)3xi = i (7)4xi = f (11)5x^1 = :^ 

(4)2xi = f (8)2x4 = f (12)2x^-:iV 

In multiplying these fractions by the whole numbers what 
change was made in the numerator to get the product? 

Make a rule for multiplying a fraction by a whole number. 



158 BATIOKAL 6RAMMAB SCHOOL ARITHMETIC 

9. Apply your rule to these problems, reducing to whole or 
mixed numbers all the products that are improper fractions: 

(1) 3xi= (5) llxA= (9) 6x 3 = 

(2) 5xf= (6) 9x,V= m 8xA = 

(3) 7 X f = (7) 4 x f = (11) 10 X I = 

(4) 9 X { = (8) 6 X } = (12) 9 X IJ = 

Note. — In such as (7) use cancellatiou, thus: i X «-= -^ = 3J. 

3 

10. Solve these problems and make a rule for multiplying 
a fraction quickly by its denominator: 

(1) 6 X i = ? (5) 12 X tV = ? (9) 75 X 4f = ? 

(2) 9xi = ? (6) 18x it =? (10) ^ X -f =? 

(3) 5 X t = ? (7) 21 X H = ? (11) wi X -^ = ? 

(4) 7xf = ? (8) 48x W=? (12) a: X -f =? 

Note.— In all such problems use cancellation, thus: /f X -s =5. 

11. Give the values of the letter in each of these problems and 

make a rule for multiplying a fraction quickly by some factor of 

7 7 
its denominator; (use cancellation thus : ? X -jj = j- = If). 

4: 

(1) 3x I =f; (3) 12x^V = f; (5) 25x^^ = f; 

(2) 5xA = f; (4) 9x/, = ^; (6) 13.x if =f 

State the rule. 

Principle VIII. — A fraction is multiplied by 

1. Multiplying its numerator by the multiplier; or, 

2, Dividing its denominator by the multiplier. 

Note. — The usefulness of the method of cancellation consists in male- 
ing these tvm processes t^ndo (balance) each other. (See problem II.) 

Query. — When will the second method (Principle VIII, 2), 
be the more convenient? 

A fraction is multiplied by its denominator by dropping Us 
denominator. 



COMMON FRACTIONS 159 

§101. Mnltiplying a Mixed Number by a Whole Number. 

1. What will be the cost of 12 yd. of muslin at 8f cents? 
Solution.— If 1 yd. costs 8f f what will 12 yd. cost? 

12X8| means 12x8+13x1 = 96 + 8 = 104. Am, $1.04. 

Note. — In dealing with such expressions as 12x8 + 12x|. which 
contain both signs, {X)and (+), the indicated multiplications should be per- 
formed first, and then the additions. 

2. 13 X 4| = ? 

13 

M 1Q N/ A Multiply the whole numbers in the usual way. 
7i = 13 X I 13 X 8 = V = 7S, as in problem 8, §100. 

59J = 13 X 4g 

PROBLEMS 

Solve the following, using the more convenient method in each 
case, employing cancellation whenever it shortens the work: 

1. 5x|i= 5. 9xfJ= 9. 2xV- = 

2. 3 X I = 6. 9 X I = 10. 3 X 6J = 

3. 5x4|= 7. 9xyV= 11. 9x3^ = 

4. 2x2i= 8. 9x f = 12. 8xy\ = 

13. Find the cost of 25 yd. of cloth at *| per yard. 

14. A earns $1|, and B earns 4 times as much. B earns $.r. 
Find X, 

15. A grocer sold J bu. of apples to one customer, and 5 times 
that quantity to another. He sold y bu. to the second customer. 
Find y. If apples were $.40 a peck what amount of money did 
the grocer receive from the first customer? from the second? 

16. If a man can cut J of a cord of wood in one day, how 
much can he cut in 6 da., working at the same rate? 

17. One boy lives -^^ mi. from school and another 4 times aa 
far. What is the distance from school to the second boy's home? 

18. Solve problems based upon the following facts: 
An iron tube weighs i lb. per foot. 

I of the weight of water is oxygen. 

The circumference of a bicycle wheel is 6J| feet. 

A west wind moved 47f mi. per hour. 



160 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



§102. Multiplying a Whole Number by a Fraction. 

The whole number is here the multiplicand. 

Definition. — To multiply a whole number by a fraction means to 
divide the multiplicand into as many equal parts as there are units in 
the denominator, and to take as man^ of these equal parts as there are 
units in the numerator of the multiplier. 

Illustration. — 12 multiplied by | means that 5 of the 6 equal parts 
of 12 are wanted. J times 12 = 2, and | times 12 = 5 X 2 = 10. 

g times 12 and | of 12 mean the same thing. 

1. Solve these problems, reading the sign (x) "times": 

(1) f X 10 = ? (4) f X 15 = ? (7) t X 45 = ? 

(2) I X 25 = ? (5) f X 28 = ? (8) y«j x 26 = ? 

(3) i X 32 = ? (6) r\ X 33 = ? (9) t\ x ^7 = ? 

Suggestion.— In (9), ^f of 47 = 4^i, and 7 X4A = 28f} = 28+lH 
= 291{. Am. 29tJ. 

2. Solve the following : 

(1) V^^ X 50 = ? (3) a X 105= ? (5) ^^ X 110 = ? 

(2) H X 39 = ? (4) 14 X 220 = ? (6) 6^ x 28 = ? 
Suggestion. — 6J x 28 means 6 X 28 + J X 28 = 168 + 21 J = 189{. 
Note.— See note to problem 1, page 159. 

§103. Factors May Be Interchanged. 

In Fig. 81. (a) the rectangle is supposed to be | in. wide and 
5 in. high. The area is then 5 times | sq. in. = 2^ square inches. 

In (b) the rectan- 



a 



gle is 5 in. long and 

^ in. high. Its area 

is i times 5 sq. in., or 

— ' " i of 5 sq. in. = 2^ sq. 

in., as is plain from 
Fig. 81. 

This shows the 
truth that when a 
fraction and a whole 
number are to be mul- 
tiplied it makes no 
difference in the nu- 
FiGUBi 81 merical value of tho 

product which of the factors we regard as the multiplicand. 

The name of the result is learned from the nature of the 
problem. 



^j 



OOMMOK FRACTIOKS 101 

Letting f- (read *'a; divided by y") stand for any fraction, and a 
stand for any whole number, state the following principle in 
symbols : 

The product of a tvhole number by a fraction^ or of a fraction 
hy a whole number ^ equals the product of the whole number by 
the numerator^ divided by the denominator, 

PROBLEMS 

1. Find the cost of 2 J yd. of cloth at $3. 

2. A man had $75; he spent f of it for a bicycle, and ^ of 
the remainder for clothing. Find the amount spent, and the 
amount he had left. 

3. A gallon contains 231 cu. in. How many cubic inches in 
f gal.? in I gallon? 

4. A cubic foot of granite weighs 170 lb. How many 
pounds in 5f cubic feet? 

5. y\ of a plot of ground containing 918 sq. rd. was fenced for 
a garden. The garden contained how many square rods? 

6. Make and solve problems, based upon facts obtained by 
yourself, or upon those given below. 

An avoirdupois pound of gold is worth about $348f . 

A bale of cotton ordinarily weighs 450 lb. Price 17^^ per 
pound. 

A tank contains 126 gal. linseed oil. Price $.62 per gallon. 

Corn is quoted and sold @ 50|^ a bu. Wheat is quoted and 
sold @ 77t^ a bushel. Oats are quoted and sold @ 32f ^ a bushel. 

To plow an acre with a plow cutting a furrow 10" wide (a 10" 
plow) a horse must walk 9y\ mi. ; with a 12"-plow, a horse must 
walk Si mi. ; with a 15"-plow, 6f miles. 

For a team drawing a plow, a distance of 16 mi. is a fair day's 
work and a distance of 18 mi. is a large day's work. The follow- 
ing table shows to what fractional part of an acre a furrow ^ mi. 
long and of the stated widths is equivalent: 

Width OP Furrow Equivalent 

12" A A. 

13" iA A. 

14" <fy A. 

15" ft A. 



Width op Furrow 


Equivalent 


16" 


irVA. 


17" 


-iVa A. 


18" 


A A. 


20" 


i«A 



1G2 



RATIONAL GRAMMAH SCHOOL ARITHMETIC 





































(i) 








U04. Multiplying a Fraction by a Fraction. 

1. What is the area of a rectangle 
3"x 5", Fig. 82 (1)? 6"x 8"? T'x 8'? 2 rd. 
X 80 rd.? 16x20? a x J? 

2. How many square feet in the area 
of a rectangle f x 8'? f 'x 12'? f x 16'? 
VV'x 30'? (Use cancellation. ) 

3. What is the area of a rectangle 
f'xf'? [See Fig. 82 (2).] 

Solution.— -.4 J5Ci) is a rectangle J" long 
and I" wide. 

AEOH is a square inch. 
What part of a square inch is ABKHI 
What part of ABKH is ABCD ? Point out 
on the figure the part that represents } of | of 
1 square inch? 

Into how many small rectangles, such as 
a, do the dotted lines divide the square inch? 
What part of the square inch is one of the small rectangles, a? 
How many of these are there in the given rectangle ABCD ? 
The area of ABCD is then what part of a square inch? 

We may write the two expressions for the area equal, thus: 

i X 4 = A. 

We see then that, if our law for finding the area of the rectangle, 
which we have found to hold for integers, is to hold for fractions also, 
we must define both i X 4 and f X | to be the same as | of }. Each is 
equal to ^s' 

It is clear that, whatever the fractions to be multiplied may 
be, the square unit [sq. in. in Fig. 82 (2)] may be divided by the 
cross lines into as many equal parts as there are units in the prod- 
uct of the denominators, and that there will be as many of these 
equal parts in the given rectangle as there are units in the prod- 
uct of the numerators. 

Hence our fractions may be multiplied by merely writing the 
product of the numerators over the product of the denominators 
with the dividing line between. Cancellation should be used 
whenever possible. 

4. Draw a square inch and find these products as above: 

(1) t" X f " = ? (4) I" X f " = ? (7) V' X r ^ ? 

(2) f X i" = ? (5) r X f " = ? (8) f X f " = ? 

(3) r X f " = ? (6) r X f " = ? (9) f " X i" = ? 



COMMON FRACTIONS 163 

5. By drawing and dividing a square inch, as in Fig, 81, show 
that the following equations are true : 

(l)4xA = 4ofA; (3) fx,«, = Aof4; (5)4xA = H 
(2),«ix 4 =Aof 4; (4) Ax? = f of A; (C) A><4 = H 
To find \ of 7^, we take a fractional unit which is \ of -f:^ 
(the fractional unit of 7^), and use the same number (5) of them, 
thus obtaining VV- B«t ^ of VV = 6 x 4 of VV = ^ x ^\ = 4^. As 
4 X i^r = f ^f V^T> 4 ^ iS = f T' Similar reasoning would show also 
that Vt ^ T = f ^- Now we recall that the numerator (30) of the 
product (rfxf) was obtained by multiplying the numerators 
(6 and 5) of the factors {^ and 4). How was the denominator 
(77) of the product obtained? 

6. State a rule for quickly multiplying a fraction by a fraction. 

7. Solve* these problems as rapidly as you can work accurately: 

(l)'fxi (5)fxi (9)ixf (13) |xf 

(2)ixi (6)ixf (lO)txt (14)fx| 

(3)fx| (7) f xf (ll)fx| (15) |xf 

(4) |xf (8) 4xJ (12) 4x1 (10) |xf 

8. Letting -| and y denote any two fractions, state the fol- 
lowing principle in symbols : 

77ie product of any two fractions is a fraction whose numer- 
ator is the product of the numerators and ivhose denominator is 
the product of the denominators. 

§105. Multiplying a Mixed Number by a Mixed Number. 

1. How many square feet in the area 
of a rectangle 4J" x 5^"? 

(1) How long is a (Fig. 83 ) ? how wide? 
What is its area? 

(2) Answer similar questions for J, 
(?, and <Z.. 

(3) How long is the entire rectangle? 
how wide? What is its area? figure 83 

Since the area of the entire rectang:le equals the sum of the areas of 
its parts we may write (see note, problem 1, page 159): 

4jX5i = 4x5+}X5 + 4X J + JX i = 20+ 3| + 2 + J = 261. 

A71S. 26t sq. ft. 

* Cancel whenever it is possible. 



^4 


b d 


4 


a c 




5 Vz 



104 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

Point out on figure the areas that represent all the parts of the 
product. 

2. Find the cost of 9^ T. hard coal at $7J. 

First Solution.— If 1 T. costs §7|, 9jt T. wiU cost 9^ x $7f. 



Convenient Form 


Explanation 


7} 


91 X 7} means 9 X 7} + 1 X 71 
i X 7| means i of 7}. 


63 =9 X7 
6} = 9 X J = Y 

3|= 1X7 
1= 4X 5 


9 X7f = 69} 
iX7J= 3J 

9iX7J = 73| 


73| 


Ana. |73|. 



Second Solution.— 91 T. = V T. ; 17? = IV 

Y X V = «!•. or 731. Ans. $73|. 

Find areas of surfaces having the following dimensions; 

3. 44^ ft. X 28f feet. 

4. 36| ft. X 27f feet. 

5. 24f ft. X 18| feet. 

6. 45f ft. X 30J feet. 

7. 27i ft. X 18f feet. 

Find the cost of the following items: 

8. 24f doz. eggs @ $.16|. 

9. 46Jgal. of oil @ $.12i. 

10. 15i yd. of cloth @ $.66f . 

11. 52i Ih. of sugar @ |.05|. 

12. 265f M. of pine flooring @ $35^. 

Make original problems from the following items: 

A cubic inch of water weighs 252f f grains. 

A water tower is 2|J ft. higher than the mound upon which 
it is built. The mound is 81| ft. high. 

G9^ statute miles = 1 degree of longitude at the equator. 

A pine tree 87 ft. high has no branches for ^f of its 
height. 

1^ of a library containing 55,447 volumes was destroyed by 
fire. 



COMMON FRACTIONS 165 

Flaxseed cost $1| per pound when a certain linseed oil factory 
bought supplies. 

A spring furnishes 28^\ bbl. of water daily. 

A certain vessel sails 11 J mi. per hour, on an average. 

A room is 32 ft. long and 24| ft. wide. Painting costs $^ per 
square foot. 

A cubic foot of water weighs 62.5 pounds. 

Gold is 19i times as heavy as water. 

30^ sq. yd. = 1 square rod. 

§106. Dividing a Fraction by a Whole Number. 

ORAL WORK 

1. What is i of 40A.? 40A. divided by 8 equals what? 

2. \ of 35 lb. = ? 35 lb. + 7 = ? 

3. tV of 80 = ? 80 + 10 = ? 

4. i of 3 fourths = ? 3 fourths + 3 = ? 

5. i of f A. = ? f A. ^ 5 = ? 

6. J of 4 ft. = ? 4 ft. -4- 3 = ? 

7. iot fjin. = ? fjin. -*-9= ? 

8. Compare these fractional units, or unit fractions: 

(1) i is what part of i? (7) ^V is what part of i? 

(2) I is what part of i? (8) j\ is what part of J? 

(3) I is what part of i? (9) J is what part of I? 

(4) tV 18 what part of i? (10) yV is what part of i? 

(5) I is what part of i? (11) xV is what part of i? 

(6) i is what part of i? (12) -g^ is what part of i? 

9. How is the size of a unit fraction changed by multiplying 
its denominator by 2? by 3? by 4? by 10? by 100? by a? by m? 

10. By what must you multiply the first fraction in each of 
these pairs to get the second? 

(1) I and I? (4) V-i and « ? (7) ^\\ and |?? 

(2) ^ and i? (5) || and jj? (8) rk and f? 

(3) A and f? (6) |J and ff? (9) j^ and /,? 

11. Make a rale for dividing a fraction by 3 without changing 
^tt numerator; by 5; by 28; by a. 



1G6 RATIONAL GBAMMAE SCHOOL ARITHMETIC 

WRITTEN WORK 

1. What nnmber should stand in place of x in these examples? 

(1) i?-. 3 = /^; (4) V/-- 16 = t\; (7) ff*9 = fT; 

(2) -V * 15 = ? ; (5) -VV - 25 = j%; (8) ¥ H.a= I- ; 

(3)W-14 = fi; (6)41^- 144 = fT; (9)^*c=^. 

a. Solve these problems by finding what number should stand 
in place of x : 

(1) 1*3 = 4; (3) H- 8 = il; (5) 4H* 3 = l|i; 

(2) 1*4 = ^; (4) V*25 = -f^; (6) |H-15=ii!-. 

3. In what two ways may a fraction be divided by a whole 
number? When would you use the method of problem 1? of 
problem 2? 

Principle IX. — A fraction may he divided hy a tvliole num* 
heVj (1) by dividing its numerator hy the wliole numher^ without 
changing its denominator^ or (2) hy multiplying its denominator 
by the whole number^ without changing its numerator. 

Note.— The note after Principle VIII, page 158, applies here alaa 

PROBLEMS 

1. Divide f lb. of shot equally among 5 boys, and give the 
weight of one share. 

2. i lb. of maple sugar is to be equally divided among 3 chil- 
dren. What part of a pound will each receive? 

3. f of a piece of work can be done in 8 da. What part of 
it can be done in 1 day? 

4. Make and solve problems based upon the following items, 
showing work in each case : 

%\\ to buy lace costing $5 a yard. 

$iJ to buy apples costing $3 per barrel. 

\ of an estate divided among 4 heirs. 

5. If f lb. of coffee is divided into 4 equal portions, how much 
is there for each portion? 

6. 8 lb. of butter cost If; lib. cost how much? 



COMMON FRACTIONS 16? 

7. A strip of tin-foil | ft. long is to be cut across into 
4 equal pieces. What is the length of one strip? 

8. 4 children have a garden containing f A. If the children 
receive equal portions, what part of an acre does each 
receive? 

9. 4 men own f of an estate, equally. What part belongs 
to each man? If the estate is valued at $3G,000, how much 
money is represented by each part? 

10. If J^P A. is divided equally among 7 men, how many acres 
are there for each man? 

§107. Dividing a Mixed Number by a Whole Number. 

1. 7 strips of carpeting cover a room S^^yd. wide. What is 
the width of one strip? 

5i yd. = V yard. 

y yd. -*- 7 = i yd., the width of one strip. 

6i is what kind of number? 

Before dividing what change was made? How was the division 
performed? 

2. A boy earned $4^ in 6 da. What did his earnings average 
per day? 

2 
How was the dividing done in this case? 

3. Make a rule for dividing a mixed by a whole number. 

To divide a mixed mimber by a whole number first change the 
mixed number to an improper fraction and then proceed as in the 
division of a fraction by a whole mimber, 

PROBLEMS 

!• 86f yd. of cloth were made up into 6 ladies' suits. If the 
suits contain the same number of yards, how many yards are 
there in each suit? 



168 RATIONAL ORAMMAR SCHOOL ARlTHMfiTIC 

2. A ceiling 16 ft. wide contains 233^ sq. ft. Find the 
length. 

233i ft. =^^ feet. 
175 

" ^"^ ■*■ ^^ = 3^1^" '^^^ " ^^^^' -^^'"^ " ^^' '^" ^^^^^ ^® ^*^® length 
4 
of the room. 

3. Solve the following problems : 

(1) 14i bu. -^ 8 = (6) $12f + 5 = 

(2) 9^Vyd.- 6= (7) 45^.5 = 

(3) 161 hr. -^ 12 = (8) ^^ + 4 = 

(4) 44yV Ib.^ 9 = (9) 14^V ^ 7 = 

(5) 39f mi. ^ 7 = (10) 66f -^■ 8 = 

Note. — In problems like (7) and (9), in which the integral part of the 
dividend exactly contains the divisor, divide the whole number and the 
fraction separately and add the quotients. 

4. Find the lengths of the surfaces whose areas and widths are 

given here : 

Areas Widths Lengths 

(1) 212^ sq. ft. 17 ft. 

(2) 2Gli " 14 '* 

(3) 406i " 25 '* 

(4) 164| " 13 " 
Make problems based on these items: 

5. A ship sails 246y^^ mi. in 36 days. 

6. A man paid $195i for 72 days' board. 

7. A field contains 2172J sq. rd. One side is 40J rd. long. 

8. 15 lb. of fresh salmon cost $2.73|. 

§108. Dividing Any Number by a Fraction. 

(A) Dividend a whole number. 

1. How many fib. packages can be filled from 3 lb. tea? 

(1) How many i lb. packages can be made from 1 lb.? How 
many J lb. packages can be made from 1 pound? 

(2) How many f lb. packages can then be made from 3 lb. 
tea? 



COMMON FRACTIONS 169 

Solution. — J is contained in 1, 4 times. } is contained in 1, i this 
number of times. } is then contained in 1, } times. How many times 
is it contained in 3? 

Or, since f X | = 1, f is contained in 1, | times. 

2. A boy earns at the average rate of $t per day selling papers. 
How many d3.ys did it take him to earn $12? 

(1) What effect does it have on a fraction to multiply its 
numerator? to multiply its denominator? 

(2) What effect does it have on a fraction to multiply both its 
numerator and its denominator by the same number? 

(B) Dividend a fraction. 

3. Divide f hy 4. 
Solution.— 

4x7 4x 7 

|=47|'^*^y^ [Seeprob.2(2)above.] i^ = ^;why? 
7X5 7X5 

(C) Dividend and divisor both mixed numbers. 

4. I paid $49| for coal @ $7| per ton. How many tons did I 
buy? 



Solution.— 



248 31 -i^ 

49J=^; 7J=^; 49|H-7f=-^ 

8 



^8 -|-X?X4 

_A L - 3 2 _ « 2 

^ ^X 5 X ^ 



Ans. 6| T. 



Let US now seek a general method of dividing any number by a 
fraction. 

5. How many times is each of these fractions contained in 1? 



170 RATIOKTAL GRAMMAR SCHOOL ARITHMETIC 

6. How many times is each of these fractions contained in 1? 

t; I; 4; i; I; tV; tV; if; ii> H> tW> -s- 

7. What number should stand in place of the letter x in each 
of these products? 

(1) fx| = a;; (4) Jx f =x; (7)T»jxV=a;; 

(2) fxi = aj; (5)T»,xV- = a;; (8) -| x ± =a;; 

(3) fx| = x; (6)|fxH = a;; (9) ^ x 4 =a:. 

8. What number should stand in place of the letter y in these 
products? 

(1) |xy = l; (3) ,Vxy = l; (5) ffxy = l; 

(2) fxy = l; (4) Jfxy = l; (6) ■^xy = l; 

1 divided by | equals f , and 1 + |=|; l->-8 = i and 1 + ^ = 8 and 
so with other numbers. 

Definition. — Dividing 1 by any number, whole or fractional, is called 
inverting the number. 

The reciprocal of any number is the number inverted. 

9. To what number is the product of any number and its 
reciprocal equal? 

10. Examine your answers to problem 7 and state how to find 
quickly the number of times any fraction is contained in 1. 

11. If a fraction is contained in one | times, how many times is 
it contained in 2? in 6? in 12? in i? in }? in i? in any number? 
in a? 

Note.— 7 may be written f. 12 may be written Vi ^J^d so on. Con- 
sequently 7 and f are reciprocals, as are also 12 and ^^ ; 25 and A, and 
soon. 

12. What is actually done by inverting the divisor in a 
problem in division of fractions? (See Definition above.) 

13. If, then, the reciprocal of a fraction is multiplied by 6, 
what operation is performed? 

14. Can you make a rule for quickly dividing any whole num- 
ber by a fraction? 

Principle X. — The reciprocal of any fraction shows how 
many times the fraction is contained ml. ' 

Principle XI. — Any number ^ mixed^ integral^ or fractional^ 
may be divided by a fraction by multiplying the number by the 
reciprocal of the divisor. 



I I.I 



COMMON FRACTIONS 171 

15. Why do we invert the divisor? Why do we mnltiply the 
reciprocal of the divisor by the dividend? 

PROBLEMS 

1. f yd. of silk costs $f . Find the cost of 1 yard. 

2. At $1J a yd. how much cloth can be bought for $f ? 

3. At $6|V per hundred how much beef can be bought for 
«2i? 

4. How many meals will 4^ bu. of oats supply, if y\ bii. are 
fed at a meal? 

5. How many pieces of wire 
^ * \ i ft. long can be cut from a piece 

' ' ' ' -^ 49ift. long? 
l^-r 6. Measure If ft. by f ft. 

PiGUBB 84 - (See Fig. 84.) 

Draw lines and illustrate the following : 

7. 2i ft. + f ft. « 10. 3f ft. -H f f t. = 

8. ^ ft. + A 't- = 11- li ft. -*- i ft. = 

9. 2| ft. + 1 ft. = 12. Divide 3i yd. by i yard. 

13. Divide 16* yd. by 2i yd. ; by If yd. ; by 4^ yd. ; by 1^ yd. ; 
by f yard. 

14. Divide $6f by $tV; by $A; ^1 H\ by $J. 

15. Find the number of strips of carpeting running the long 
way of the room : 

Width of Rooms Cabpetino 

22i ft. ^ ft. wide. 

7fyd. I yd. - 

18i ft. ^ ft. ** 

8i yd. I yd. *• 

15ift. . 2i ft. *' 

Make and solve problems based upon the following items: 

16. Portland cement $2} per bbl. ; $9 purchasing fund. 

17. Sidewalk having an area of 607| sq. ft.; flagstones 
5^ ft. square. 

18. 15§ gal. maple syrup; fgal. measures. 

19. 29f doz. oranges; ^ doz. oranges. 



172 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

20. The ratio of two numbers is f; one of the numbers is f. 

21. A man bought a house for $1860. The house cost him f 
as much as he paid for a mill. What was the cost of the mill? 

22. My library contains 1075 volumes, which are f as many 
as a friend's contains. How many volumes are in my friend's 
library? 

23. Of a number of cars of grain inspected on a certain day 
86 were rejected. This number was J| of the entire number 
inspected. Find the number. 

24. Let T- and — denote two fractions, and state the foUow- 

h y 

ing principle in symbols: The quotient of one fraction by 
another is equal to the product of the dividend by the inverted 
divisor. (Compare Principle XI.) 

§109. Complex Fractions. 

Any problem in division of fractions may be written in frac- 
tional form, the dividend being written above and the divisor 
below the line. 

Example. — The problem, to divide f by ^x ^^^J be written: 

Or, the problem, to divide 18| by 6|, may be written: 

6| • 

Definition. — Fractions that contain fractions in one or both terms, 
are called complex fractions. 

Principle XI, §108, will enable us to solve all such problems. 
If either the divisor or the dividend, or both, are mixed numbers, 
they should first be reduced to improper fractions. 

Following this principle : 

^-♦^¥ = H- 

The second step becomes unnecessary if we notice that the 
product of the outside terms (6 and 11), called the extremes^ gives 
the numerator, and the product of the inside terms (7 and 5), 
called the meansy gives the denominator of the result. 



COMMON FRACTIOKS 



173 



A complex fraction is simplified when its terms are freed of 
fractions, and the resulting fraction is expressed in its simplest 
form. 

In many cases factors may be cancelled from both numerator 
and denominator before multiplying the fractions. Cancel when- 
ever possible. 

Show that the method explained is the same as dividing the 
product of the extremes by the prod act of the meaiis in the given 
complex fractions. 

PROBLEMS 

Simplify the following complex fractions, cancelling when pos- 
sible : 



1. 



3. 4- 



1 

1 



4. 



6. 



* 

4 

6| 



L 
3 

15| 

T 



10. 



11. 



12. 



13. 



84f 
62| 



14 M 



15. 



§110. Joint Effects of Forces. 



^/' 



B 



T^^r^T^ 



^??r^?u 



"^^^ 



j^ 



ISI 



Figure 85. 

1. The rear car, C, pulls back on the middle car, P, with a 
force of 100 lb., and the front car, A^ pulls it forward with a force 
of 250 lb. If car, ^, were taken out of the train whal; single for- 
ward force would draw it the same as it is being drawn in the train? 

2. A force of 23 lb. is pulling the car (Fig. 9C ) toward the 

right and another of 12 lb. is pulling it 

"^ ^iQi^^rjQL . J *' toward the left. If the car is free to move, in 

FiGURB9a which direction will it move? What single 

force would move the car in the same manner 

as do both of these forces pulling at the same time? 



174 RATIONAL GRAMMAR SCHOOL ARITHMETIC ' 

8xib. 3. What single force will produce the 



tilWnmTOfi't^ s^me motion of the car (Fig. 97) as 82; lb. 
FiGUBBQ? acting toward the right and 5x lb. acting 

at the same time toward the left? 
^^=^' r I I -rno ^^^ 4. What single force will move the car 



FIQURB08 (^^S- ^^) '^ ^^® same manner as 6m lb. 

pulling against 27n lb. will move it? 

When a force of 18 lb. is supposed to be pulling a car toward 
the rights it will be written thus : R 18 lb. When the same force 
pulls toward the lefty it will be written thus : L 18 lb. When 
two forces, R 16 lb. and L 12 lb., act at the same time, their joint 
effect is the same as the effect of the single force, R 4 lb. 

5. What would be the joint effect of R 28i lb. and L21i lb.? 

6. What would be the joint effect of each of these pairs of 
forces : 

(1) R 48i lb. and L 12f lb.? (4) R 23^ lb. and L 68 ^V lb.? 

(2) R 75yV lb. and L 69 ,V lb.? (0) R 16^58 lb. and L 81^ lb.? 

(3) R 181i lb. and L 129^^ lb.? (6) R 87f lb. and L 6841 lb.? 

A force of 25 T. (tons) , -pulling forward^ may be written F 25 T. > 
and a force of 18 T., pulling backward^ may be written B 18 T. 
The joint effect of both would be written thus : F 7 T. 

7. What is the joint effect of the three forces: F 17 T., F 12 T. 
and B 25 T.? 

8. What is the joint effect of each of the following sets of 
forces : 

(1) F 80^ T., B 28A T., and B 96f T.? 

(2) F 49i T., B 84| T., and F 248 jV T.? 

(3) B 68y\ T., F 73| T., and B 12^ T.? 

(4) F 28^% T., F 18i T., and B 37^^ T.? 

(5) F 692i T., B 386f T., and B 307^ T.? 

(6) F 2Sx T., F dx T., and B 36a; T.? 

9. A force of 182 lb. pulling upward may be written U 182 lb. 
How would you write a force of 78 lb. pulling downward? 

10. What is the joint effect of each of the following sets of 
forces : 

(1) U 8f lb., D 16^ lb., and U 3f lb.? 

(2) U 16i lb., U28 lb., and D 29^ lb.? 



COMMON FRACTIONS 1T5 

(3) D 29 oz., D 32i oz., and D 16f oz.? 

(4) D 29f oz., U 65f oz., and D 42^ oz.? 

11. If we call a force of 16 lb. pnlling forward, or upward, 
or eastward, or northward, or to the right, a positive force, and 
write it + 16 lb., we should call the same force pulling backward, 
or downward, or westward, or southward, or to the left, respe^- 
ively, a negative force. How should we write it? 

12. Give the joint effect, with proper sign (+ or — ) of each 
of these sets of forces : 

(1) 2 forces, each -f 8 J lb., and one force, — 12 J lb. 

(2) 3 forces, each 4- 12J oz., and 4 forces, each — 8 J oz. 

(3) 5 forces, each - 16f T., and 12 forces, each + 9 J T. 

(4) 18 forces of - 16^^ lb. each, and 20 forces of + 25^ lb. each. 

§111. Exercises for Practice. 

1. Find the following sums and differences as rapidly as you 
can work correctly. 

(1)4 + Vt = ? (7)8U4^=? 

(2)i|-i = ? (8)94-3|J = ? 

(3) H-A = ? (9) 16i-10^ = ? 

(4) if + il=? • (10) 210VV + 49VV=? 
(5)!|-il=? (11) 37i-18/T = ? 
(6) H + IJ = ? (12) 10U-98tV = ? 

2. Find the products and quotients as rapidly as you can work 
correctly, using cancellation when it makes the work easier. 

(1) ixH = ? (9) 18f x9f = ? 

(2)Ax|f = ? (10)18^*161=? 

(3)HxH = ? (11) 21f*U^ = ? 

(4)U-/t = ? (12)6*15=? 

(5)il-A = ? (13) 8i| * 124 - ? 

(6) ^f--,»ir = ? (U) 154*3i = ? 

(7) l|xA = ? (15) U*6| = ? 

(8) 8f + 3| = ? (16) 261i + 80S = ? 

3. Find the simplest results for these exercises: 
(l)ix|xT«r-? (3) (Vnx8i)H.6i-? 
(8)*Xt«^xH-? (4) (94x4f)*3i-? 



(5) ,»,+ 


tV + H = ? 


(6) * + *-« = ? 


(7) A X 


TVxif = ? 


(8) 1*1 + 


tV + H = ? 


(9) tS- 


f + f = ? 


(10) m 


-3J + 6i=? 



176 BATIOKAL GRAMMAR SCHOOL ARITHMETIC 

(11) I6f-h70f-87i = ? 

(12) 16| X 3t X If = ? 

(13) (9f^6i)x6| = ? 
Note— First divide 9f by 61 and 

then multiply the quotient by 6§. 

(14) (224 ■*• Hi) X 19U = ? 

(15) (18i + 27f)+24t = ? 
§112. Dividing Lines, and Angles. 

Problem I. — Divide the line AB into 2 equal parts, see 
Problem VI, p. 101. 

EXERCISES 

1. Draw a straight line with a ruler on the blackboard and 
Disect it with crayon, string and ruler. 

2. Bisect each of the 3 sides of a triangle and connect each 
mid-point with the opposite corner of the triangle. These. lines 
are the medians of the triangle. How do they cross each other? 

3. How might a line be divided into 4 equal parts by repeating 
this method? Divide a line into 4 equal parts by this method. 

Problem II. — Divide the angle BAD into 2 equal parts. 

Explanation. — With any convenient 
radius and with the pin foot on A {vertex) 
draw arcs 1 and 2 across AB and AD. 

Place the pin foot on the crossing point at 
2, and with a radius longer than half way from 
2 to 1 draw an arc 3. 

Now place the pin foot on 1 and with the 
radius used for arc 3 draw arc 4 across arc 3. 
» , «i .J, Call the point of crossing C 

Angle Bisected ^j^j^ ^j^^ ^^^^ ^^^^ ^^^ bisector AC. 

PiGUBB 09 Then angle BAC = angle CAD, 

EXERCISES 

1. Draw an angle on paper, or on the blackboard, and with 
pencil, or crayon and string, bisect the angle. 

2. Draw a triangle and bisect each of its 3 angles. How do 
the bisectors of the angles cross each other? 

3. The line CD of Fig. 35, p. 101, drawn from C to D, 
is called the perpendicular bisector of the line AB. Draw a 
triangle of 3 unequal sides, and then draw the 3 perpencMcular 
bisectors of its sides. How do these lines cross each other? 




COMMOK FKACTIOKS 




§113. The Parallel Ruler. — A very good parallel ruler may be made 

by catting out two strips of cardboard or of very thin wood, exactly 

alike, as shown at (a), 

Fig.lOO. Thenumbeni 

show the widths and 

the lengths. To use 

the ruler place the 

two parts with their 

Fong sides together, 

the thin ends being 

in opposite directions 

as in the cut. 

Another sort, which 
the pupil may make 
for himself or pur- 
chase for a few cents, 
is shown at {h) , Fig. 100. The strips 8 may be of light cardboard or 
of thin wood of the lengths and the widths shown in the cut. The 
outside edges of these strips should be made as smooth and as 
straight as possible. The cross strips, which may be narrower than 
the strips Sy should be of exactly the same length between the 
pins at Cy Z>, and at E^ P. The distances EC and PD between 
the pins, should also be made equal. Common pins may be stuck 

through and bent over 

^ W _ to hold the strips to- 

**-• gether at (7, 2), E^ and 

^ _ ' ~ P. 

^' " "'*"^ Z The pupil should 

provide himself with 
a parallel ruler for the 
following problems : 



FiGUBB 100 




(h> 




FiGUBB 101 



Problem I. — Draw 
a line parallel to a 
given line. 

Explanation. — Let 
AB be the given line. 

Fig. 101 (a) shows how 
this is done with the first 
kind of ruler, by holding 
the part 2 on the line, 
and sliding part 1 along, 
drawing the lines along 
the uppei edge of part 1. 



Fig. 101 (6) shows how to solve the problem with the second kind ol 
ruler. 




178 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

Problem II. — Draw a line parallel to a given line and through 
a given point. 

Explanation. — Sup- 

X. 2) P?s© ^^ (Fig. 102) is the 

given line, and P the 
point. 

Hold one edge of the 
parallel ruler along AB^ 
raise the other strip until 
its edge goes through 
the point P, and draw a 
line CD along this edge. 
PiGUM 102 CD is the desired line 

Problem III. — Solve Problem II with ruler and compass. 

Explanation. — First Step: Place the pin foot on the given point 1* 
and spread the feet until the pencil foot reaches some point, as C, on the 
line AB, Fig. 103. Draw the arc C2. 

Second Step: Place the pin 
foot on C and using the same 
radius as before, draw arc PU 
cutting AB9XD. 

Third Step: Spread the feet of 
the comnass apart as far as from P 
to D, ana placing the pin foot on 
C, draw the short arc a. Connect 
the crossing point E with P. EP Fiouaa 108 

is the desired parallel to AB through P. 

EXERCISES 

1. Draw a line on paper, mark a point not in the line and with 
a parallel ruler draw a line through the point and parallel to the 
first line. 

2. Solve Exercise 1 on the blackboard with chalk, string, and 
ruler. 

3. Draw a triangle on the blackboard. Draw a line through 
each corner of this triangle and parallel to the opposite side. 

Problem IV, — Divide a line AB into 3 equal parts (or trisect 
AB). 

Explanation. — Draw an indefinite line 
ACy making amr convenient angle with AB, 
Measure off 3 equal spaces from A to- 
ward O. Connect D with B. 

Draw through the points 2 and 1 lines 
parallel to T>B (see Problem II). Then AE^ 
Line Trisected ^^ = ^^' *^^ ^ ^"^^ ^ *''® **^® triseotioD 

Figure IM points. 





COMMON FRACT10K8 179 

EXERCISES 

1. Draw a line on the blackboard and trisect it. 

Note.— Draw the parallels by the method of Exercise 3 under Prob- 
lem III above. 

2. Prom the suggestion of Fig. 105 draw a line _^.^C 
on the blackboard and divide it into 5 equal a*=*=^^-^^^-^B 

parts. FiGTJM 105 

3. How may a line be divided into 7, or 9, or 13 equal parts? 
Draw a line on the blackboard and divide it into 7 equal parts. 

4. How does the line connecting the opposite 
corners of a square divide the square? 

5. Answer a question like 4 for the rectangle ; 
for the parallelogram. 

6. If, then, the area of a square equals the 
FiouBs 106 product of its base and its altitude, what is the 

area of one of the 2 equal triangles into which a diagonal divides 
the square? 




Rect-dngle 
III bflag 



i?7 ^r^Ti^ 



Figure 107 

7. Answer similar questions for the rectangle; for the parallel- 
ogram. 

8. The area (A) of a rectangle of base h ft. and altitude a ft. is 
how many squarQ feet? 

9. What is the area of a right-angled triangle (a right 
triangle) of base h in. and altitude a inches? 

10. The base of a parallelogram is h in. and the altitude is 
a in. ; what is the area? 

11. The base of a triangle is h and the altitude is a; what 
is the area? 

12. The bases of a rectangle, of a parallelogram and of a tri- 
angle are b in. and their altitudes are a in. Find the ratio of the 
area of the rectangle to the area of the parallelogram ; the ratio of 
the area of the parallelogram to the area of the triangle. 



180 



RATIONAL ORAMMAB SCHOOL ARITHMETIC 



^114. Vm8 of the 30'' and the 46'' Triangles. 

Pboblsm V. — To make the triangles for use in drawing. 



Explanation. — (a) Fold a piece of smooth heavv paper, having one 
straight edge (like the piece shown in Fig. 108), over a line near the middle. 
Bring the straight edges carefully together as shown in Fig. 109. 





FIOUBB 106 



FlGUBB lOQ 



Crease the paper smoothly with a ruler or a paper knife and paste or 
glue the two pieces together. When the paper is dry, mark off distances 
of 4" from the square comer on the crease and on the straight side. 
Connect the 4 in. marks and cut the paper smoothly along the connecting 
line. This will give a triangle of the form T, Fig. 110. 

(b) In the same way, fold, crease, and paste another piece of paper 
a little larger than before. On the straight side mark off a distance 
CA equal to 3". With compasses, or with a stringer ruler, mark a point, 
B, on the crease so that AB equals 6''. Draw AB and cut out the tri- 
angle fif. Fig. 111. 




4" 

FlOUBB 110 




FlGUBB 111 



If preferred the triangles T and S may be made of thin wood. 

Exercise.— Place the side, AC, of S against the long side of T, and 
holding the triangles witli the left hand, draw a line along AB. Now 
hold T, slide S a little (say \") and draw another line along AB. Simi* 
larly, draw a third line. Lines in such positions are called paraiW. 
lines. 



OOMIION PRACTI0N8 



181 



Problem VI. — Through a given point with a triangle draw a 
line parallel to a given line. 

Explanation.— AB is 
the given line and P is the 
point the parallel is to pass 
through. 

Placje a ruler CD, Fig 
112, in such a position that 
when the triangle S is 
placed against it, one of 
the sides of 8 will lie 
along AB. Press the ruler 
against the paper and hold 
it with the left hand; 
with the right, slide the 
triangle alon^ the ruler 
until its side just touches 
the point P. Draw line 
FE through P and along 
the edge of the triangle. 
FE is ^the desired parsdlel 
line. 




FiGUBB 112 




FIGUBB 113 

Fig. 113 shows how this problem is solved with the triangles alone. 
EXERCISES 

1. Solve the first exercise of Problem III with the triangles 8 
and T as shown in Fig. 113. 

2. Draw a triangle with sides of 1", IJ", and 2", and through 
each corner draw a line parallel to the opposite side. Use the 
ruler and the triangles. To draw the triangle see Problem IX, 
p. 103. 



182 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

Problem VII. — At a given point on a line with the triangles 
draw a perpendicular to the line. 

Explanation.— Let AB be the line and let P be the given point. 





FlGUBB 114 FIGURB 115 

Holid one of the triangles, as T, in the position shown in Fig. 114, and^ 
placing one side of the other triangle, 5, against the upper side of T, slip 
8 along until the square corner conies to the point P. Hold S firmly and 
draw the line PD along the side of S. 

PD is the required perpendicular. 

Problem VIII. — Through a given point not in a line, with the 
triangles draw a perpendicular to the line. 

Explanation.— Let AB be the line and let P (Fig. 115) be the given 
point. 

Hold one of the triangles, as T, so that its side lies along AB and slide 
the other triangle, S, with its side against the upper side of T until it 
oomes up to P. Then draw PD. 

PD IS the required perpendicular. 

EXERCISES 

1. Draw a line, mark two points on it 1" apart and draw a per- 
pendicular to the line at each of the two points (by Problem VII) . 
Mark a point on one of the perpendiculars 1" above the given line 
and at this point draw a third perpendicular completing a 1" 
square. 

2. Draw a triangle and through each corner draw a perpendic- 
ular to the opposite side (by Problem VIII). How do these per- 
pendiculars cross? 

3. Draw any circle, also a diameter, marking its ends A and B. 
Through its center draw a perpendicular radius and prolong it, 
making a diameter, CD. Connect AC, CB, BD and DA^ forming 
an inscribed square. 



COMMON FRACTIONS 



183 



Scaler: 6 




Scate r:20' 



§115. Scale Drawings of Familiar Objects. 

1. Notice the scale of the drawing of the 
playhouse (Fig. 116), and compute the lengths 
of the following dimensions : 

(1) The width; (2) the height of the lower 
side; of the higher side; (3) the rise (dif- 
ference of higher and lower sides); (4) the 
height of the door; the width ; (5) the distance 
from the right side of the door to the right 
corner of the house. 

2. From the dimensions in Fig. 116, with ruler and triangles, 
make a drawing of the playhouse to a scale 4 times as large as 
that of the drawing (Fig. 116). 

3. From the scale of the drawing (Fig. 
117) find the following dimensions of the 
house : 

(1) Tbe width; (2) the height of the 
eaves ; (3) the rise {ah) ; (4) the width and 
the height .of the door; (5) the width and 
the height of the window. 

4. Find the area of -the end of the house, including the gable 
and excluding the areas of the door and the window. 

5. Make an enlarged drawing of the house to a scale B times 
as large as the scale of the drawing (Fig. 117). 

6. From the scale of Fig. 118, 
give the length and width of the door; 
the length and the width of the panel. 
(Find the length by measurement). 

7. Similarly, give the length and 
the width of the door B, also the length 
and the width of the upper panels, 
and the widths of the strips enclosing 
the panels. 

8. Make an enlarged drawing of 
each of the doors to a scale 5 times as large as the scale of the 
drawings (Fig. 118). 




FiGUBB IIV 




184 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 





9. HofV^ long are the cross-pieces and the strings of the kite 
represented by the drawing (Fig. 119)? 

10. Make an enlargec? drawing of the tite to a 
scale 8 times as large as that of Fig. 119. 

11. Notice that the horizontal piece divides the 
surface of the kite into two trapezoids. The alti- 
tude of the upper trapezoid in the drawing (Fig. 
119) is li'\ *^^ ^^*^ of ^^^ lower trapezoid is fj". 
How many square feet of paper are needed to cover the kite? 
(Allow 144 sq. in. for folding and pasting over strings.) 

12. What is the length of each leg of the 
chair shown in the drawing (Fig. 120)? What 
is the height of the back? the depth of the 
seat to the extreme rear? 

13. Make a drawing of the chair to a scale 5 
times as largo as that of Fig. 120. 

14. Find from the drawing of the gate (Fig. 
121), (1) the height of the high end of the gate; 

Sc*h n40^ ^ (2) of the lower end ; (3) the 

FiGUBB 120 length of the long brace; 
(4) the length of the gate; (5) the width of 
the strips. 

15. Make a drawing of the gate to a 
scale 8 times as large as that of Fig. 121. iU»"- V6' 

16. Fig. 122 is a scale drawing of the figurb 121 
side and the end views of a large book. How long 
is the book? how wide? hovr thick? 

17. Make an enlarged drawing of the book to a 
scale 4 times as large as that of Fig. 122. 

18. From your own measurements make a 
drawing, to any convenient scale, of a thick object, 
as a block, a brick, a crayon-box, showing two 
views as in Fig. 122. 

19. Make a scale drawing from your own 
measures of a desk, table, bookcase, or other 

object in your schoolroom, showing three different views (top, 
side, and edge views) of it. 





KHo View* 

Figure 123 



COMMON FRACTIONS 



185 



§116. SchooUiouse and Oronnds. 

1. Using a foot rule, graduated to 16ths of an inch, and 
regarding the scale of the drawing (Fig. 123), find the width of the 
grounds; the length; 
the area in square 
rods. (30^ sq. yd.= 
1 square rod.) 

2. Find the length 
of the field; the 
width; the area in 
square rods. 

3. Find the length 
and the width of the 
school yard ; the area 
in square rods. 

4. Find the length 
and the width of the 
schoolhouse ; the area, 
in square yards, cov- 
ered by it. 

5. How far is the 
front door of the 
schoolhouse from 
the front fence? from 
the west front gate? from the sand pile? from the tree? 

6. How far is it from the back door to the east flower bed? to 
the back fence? to the west fence? to the coal shed? to the north- 
east corner of the school yard? to the south end of the pond? to 
the hiU? to the nearest point on the creek bank? to the foot 
bridge (F)? 

7. How wide is the south road? the creek? the branch? 

8. How many square rods in the south road in front of the 
grounds? in the crossing of the roads? 

9. How many square rods in the meadow? in the grove? in the 
pasture? 

10. How many square rods are covered by the creek and the 
branch together, within the fence lines? 




ROAD 



1 r 



Settle n 160' 



Figure 123 



186 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

11. How many rods of fence will be needed to encloBS the 
grounds and to ran along the lines indicated? 

12. From your own measurements make a similar drawing, to 
a convenient scale, of some tract of ground (school yard or field) 
near your schoolhouse. Locate any fixed objects on your tract 
in their proper places on the drawing by measuring their shortest 
distance from a fence and from a comer. 

§117. Proportion. oral wore 

1. Compare the ratio 3 : 2 with the ratio 6 : 4. What do 
you find? 

2. Compare the ratio 8d. : 2d. with the ratio 16 men: 

4 men. 

3. Compare the ratio 10 ft. : 800 ft. with the ratio 100 mi. : 
8000 miles. 

4. A proportion is an equation of ratios. Thus 6 : 12 = 3 : 6 
and A = f *^re two different ways of writing the proportion. 

Definitions.— The first, second, third, and fourth numbers of the pro- 
portion are called the first , secondly third, and fourth terms of the propor- 
tion. The first and fourth terms are called the extremes, and the second 
and third terms are the means. The first two terms are the first couplet, 
the third and fourth terms are the second couplet, 

5. In 6 : 12 = 3 : 6, to what is the product of the extremes 
equal? the product of the means? 

6. Answer the same questions for 3 : 5 «= 12 : 20. 

7. Which of these pairs of ratios may form proportions: 

4 : 9 and 8 : 18? 6 : 11 and 18 : 33? 1:3 and 8 : 21? 
3 : 7 and 12 : 28? 6 : 11 and 12 : 22? 1 and j\^? 

SL and ,^? « and S.? ^ and Jg-? 

la 2b ax e» n 6n 

8. Is 4 = i\ a proportion? 

Multiply both sides by 21 and we have 4 x 21 = 15. 

Now multiply both sides of this equation by 7 and we have 

5 X 21 = 7 X 15. 

What were the terms 5 and 21 in the proportion called? What 
were the 7 and 15 in the proportion caiiti? 



RATIO AND PKOPORTION 



187 



9. Compare the product of the Ist and the 4th numbers in 
these proportions with the product of the 2d and the 3d 
terms: 

(1) 4 = il ; (5) t\ = M ; 

(2)A=i?; (6) i-U.'^ 

(3)H= fl; (7) -^«H-;; 

Wl =V/; (8) f = *-*f 

Can you state the principle problems 8 and 9 illustrate? 

Peinciple. — In a proportion the product of the means equals 
the product of the extremes. 



WRITTEN WORK 



1. What must x be in each of these expressions to give a pro- 
portion? Solution of first equation: 2x = 15, or a; = 7.5. 



1. 2:3»5:a; 

2. 4:3 = 8:a; 

3. 6: a; = 9: 27 

4. 7:2 = a;:14 

5. a;: 6 = 8: 12 



6. a:b = 2a:x 

7. 18: a; = 54: 18 

8. 4:6= a;: 9 



10. ?* = i 

bo e 



2. What does the letter stand for in each of these proportions? 



(1) - = - 

^''' 7 85 
(4)1 = 1? 

^ ' 11 TO 

W 7-30 
11-1 

a 12 
o^ 16 



(6) 

^'^ 10 "45 



(«)il 



°36 



(Q^ a: _ 36 . 
V"'' 18 "324' 

5 13. 



'66' 



(11) , 

11- ^. 

44" 8' 



— - 



(12) 

^'^^f 161-14' 



(16) 



m' 

15 _£. 

18 ~ 4' 



188 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

§118. Practical Applications. 

1. At the rate of 20 lb. for $1.00, what will 15 lb. of sugar cost? 

2. A boy buys 36 oranges at the rate of 3 for 5 cents. What 
do they all cost him? 

Note. —Write the problem in the form f = — and find c, the cost of 
the'oranges. Use Principle § 1 1 7. 

3. The boy sells his 36 oranges at the rate of 2 for 5 cents. 
How much does he receive for them? 

4. A man walks 54 miles at the rate of 9 mi. in 2 hr. How 
long does it take him to walk the 54 miles? 

5. A number of boys buy 72 marbles at the rate of 6 marbles 
for 5 cents. How much do the 72 marbles cost? 

6. If it takes 5 yd. of ribbon to trim 4 hats, how many hats 
can be trimmed with 37^ yards? 

7. At the rate of $4.00 for 5 dozen, what will 3 doz. cocoanuts 
cost? 

8. In a class there are 3 girls to every 4 boys. If there are 24 
boys in the class, how many girls are tl^ere? 

9. At the rate of $7.00 for 3 days, how much money does a 
man earn in 21 days? in 27 days? in 16^ days? 

10. If it takes 7 yards of gingham to make 4 aprons, how 
many aprons will 42 yd. gingham make? 56 yd.? 594- yd.? 

11. If f of a ton of coal is worth $4.81, what are 12 T. worth 
at the same rate? 

Note. — Call the unknown term x. Write the proportion, using a?, 
and then use the^Principle, §117, to find x. 

12. A spelling class of 52 pupils writes a total of 1352 words; 
at the same rate how large a class will write a total of 728 words? 

13. A girl jumping a rope makes 441 skips in 3 min. How 
long will it take her to make 392 skips at the same rate? 

14. If the average column in a newspaper contains 1600 words, 
how much should a writer receive for 700 words at the rate of 
$5 per column? 

15. In going 10,725 ft. the front wheel of a bicycle revolves 
1430 times. How far would it go in making 1001 revolutions? 



RATIO AND PROPORTION 



189 



_6£ECh eftOOK, 




16. A clock ticks 7 times in 5 sec. ; how many times does 
it tick in 2 da. 6 hours? 

17. If 60 A. cost $3000, how many acres will cost $2450? 

18. The line BC (Fig. 124) 
represents 20 rods. FE is twice 
as long, AF four times as long, 
CD and ED iBve times as long, 
and AB six times as long. 
Find the length of each side. 

19. Find the value of x in the 
following proportions: 

EC :FE=AB:(x); 
BC : (x) = FE : CD; 
AF:FE= (x) : BC, 

20. When a foot rule is held 
Figure 134 2^ ft. (arm's length) in front 

of the face as shown in Figure 125, 7 in. on the ruler seems 
just to cover the edge of a door 7 ft. high. If the ruler is held 
parallel to the edge of the 
door, how far is the eye from 
the door? 

21. If the shadow cast by a 
4-ft. stake is 5 ft. long, how high 
is a tree which casts a shadow 50 ft. long on the same day and 
hour? 

22. The shadow cast by a 3^ ft. stake is 10| ft. long. How 
high is a flagpole that casts a shadow 225 ft. long at the same 
time? 

23. How high is a house that casts a shadow 120 ft. long at the 
moment when a stake 6 ft. high casts a shadow 20 ft. long? 

24. A road runs 2 miles south, then 1 mile west, then f mi. 
south) then 1^ mi. southeast, and then 2^ mi. south. In a scale 
drawing of the road the two-mile part is represented by a line 18 
in. long. How long a line will represent each of the other parts 
of the road in the same drawing? 

25. What is the scale of the drawing just mentioned? 

26. What is the scale of a drawing in which a distance of 20 ft. 





'%ft"^ 






-— "1 


^ 


f 


Figure 125 





[ 



190 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

is represented by a line 2| in. long? By a line | in. long? By a 
line 2i ft. long? 

' 27. The food parts of beef are water, fat, and protein. In 1 
lb. of loin steak there are 3| oz. of water, 2| oz. fat, 2| oz. pro- 
tein, and 2^ oz. of waste material. How much of each of these 
parts are there in 3 lb. of loin steak? In 5J lb.? In a 12-pound 
loin steak? 

28. In 4 lb. sirloin steak there are 34| oz. water, lOf oz. fat, 
lOf oz. protein, and 8y^y oz. waste matter. How much of each is 
there in an 8 -lb. sirloin steak? In 24 lb. sirloin steak? In 30 
pounds? 

29 In a 10-lb. porterhouse steak there are 5^ lb. water, If lb. 
fat, If lb. protein, and IJ lb. waste. How much of each is 
there in a 30-lb. porterhouse steak? In 15 lb.? In 18 lb.? In 
48 pounds? 

30. In 15 lb. of beef ribs there are GJ lb. water, 3} lb. fat, 2J J 
lb. protein, 3} lb. waste. How much of each is there in 10 lb. 
beef ribs? ' In 25 lb.? In 10 lb.? In 75 pounds? 

DECIMAL FRACTIONS 

§119. Notation of Decimals, oral work 

1. In $1111 what does the 1st 1 on the right stand for? the 
2d? the 3d? the 4th? 

2. In $6666 what does the 1st 6 on the left denote? the 2d? 
the 3d? the 4th? 

3. In $372.68 what is the unit of the 3 (Aiis, $100)? of the 7? 
of the 2? the 6? the 8? 

4. In $5555 how does the number denoted by each 5 compare 
with the number denoted by the 5 to its left? to its right? 

5. How do the units of the places of a number change as we 
pass through the number from left to right? 

6. Which unit is the fundamental unit out of which the other 
units are made? How is the place of this fundamental unit 
indicated in $675.28? 

Definition. — A dot, called the decimal point, or point, is used to show 
the units' digit. The point always stands just to the right of the units' 
digit or place. 



DECIMAL FRACTIONS 101 

7. If the law of problem 5 holds in 444.444, the unit of the 
let 4 to the right of the point equals what part of the unit of the 
Ist 4 to the left of the point? The unit of the 2d 4 to the right 
equals what part of the unit of the 1st 4 to the right? 

8. In any number what part of the unit of the 1st place to 
the left of the point equals the unit of the 1st place to the right? 
of the 2d place to the right? of the 3d? of the 4th? 

Definition. — The unit of the 1st place, or digit, to the right is called 
the tenth; of the 2d place, or digit, the hundredth; of the 3d, the 
thousandth; of the 4th, the ten-thousandth and so on. 

§120. Numeration of Decimals. 

1. The number 444.444 might be read, "4 hundreds, 4 tens, 
4 units and 4 tenths, 4 hundredths, 4 thousandths"; but it is 
simpler to read it, **Four hundred forty-four a7id four hundred 
forty-four thousandths.^^ Bead 234.234 both ways. Which is the 
shorter? 

2. The latter mode of reading ia the one used in practice. It 
maybe stated thus: Read the integral (whole) part of the number, 
then read the part of the number to the right of the point just 
as though it were a whole number standing to the left of the 
point, then pronounce the name of the unit of the last digit on 
the extreme right. The word '*and" must be pronounced only at 
the decimal point. Read 675.328; 236.89; 7.65; 43.6587. 

3. Read the following 

(1) .1; .01; .001; .0101; 1.1; 10.1; 

(2) .6; 6.06; 66.6; .0606; 600.06; 6000.006; 

(3) .60; 7.070; 8.50; .0600; 87.087; 10.0008. 

4. What is the ratio of 6 to .5; of .5 to .05; of .5 to .005; 
of 55 to 5.5; of 55 to .55? 

5. Find the ratio of 875 to 87.5; of 87.5 to 8.75; of 875 to 
8.75; of 875 to .875. 

6. How is a number affected by moving the decimal point 
1 place toward the left (see problems 4 and 5)? 2 places? 
3 places? 6 places? 

7. What is a quick way of dividing any number by 10? 100? 
1000? 



192 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

8. Express the ratio of 2.358 to 23.58; to 235.8; to 2358; 
to 23,580; to .2358; to .02358. 

Query. — Where is the point supposed to be in 2358? When 
the decimal point is not written, where is it supposed to be? 

9. How is a number changed by moving the decimal point 
1 place to the right? 2 places? 3 places? 6 places? 

10. Make a rule for quickly multiplying any number by 10; 
by 100; by 1000; by 1 followed by any number of zeros. 

11. Referring to problem 4, can you tell what effect is produced 
in such a number as .683 or .492 by writing a zero between the 
decimal point and its first digit? 2 zeros? any number of zeros? 

12. Compare .5 with each of these numbers: .50; .500; 
.50000. What is the effect of writing any number of zeros to 
the right of the last digit of a decimal? 

Definitions. — A decimal fraction, or decimal, is a fraction whose 
denominator is 10, 100, 1000, or some power of ten, in which the denom- 
inator is not written but is indicated by the position of the decimal point. 

A power of 10 is a number obtained by using 10 as a factor any 
number of times. 

§121. To Seduce a Decimal to a Common Fraction. 

The fractions we have studied whose denominators are actually 
written, are called Common Fractions. 

1. Express the following decimals as common fractions: 

.5; .50; .500; .4; .40; .400; .6; .60; .25; .250; .125; .375; 
.625; .875. 

2. Reduce the results of problem 1 to their lowest terms. 

3. Write these mixed decimals as improper fractions : 

1.5; 2.50; 2.75; 10.4; 12.5; 6.75; 18.25. 

4. Eeduce these improper fractions to their lowest terms. 

5. After dropping the decimal point from the following 
decimals, what numbers must be written beneath them to express 
them as common fractions : 

.6; .67; .625; .875; .1275; 12.75; 25.786; 33.333; 6.6666? 

6. Make a rule for expressing any decimal as a common frac- 
tion in its lowest terms. 



DECIMAL FRACTIONS 193 

7. How may 2| tenths be expressed as a decimal? 62^ han- 
dredths? 33i hundredths? 14 and 666^ thousandths? 

A pure decimal is a decimal whose value is less than 1 ; as, 
38 thousandths, QQ^ hundredths. A mixed decimal is a decimal 
whose value is greater than 1: as, 3.58 or 2.87|. 

8. Read and give the meaning of these mixed decimals: 
.12i; 3.33i; 18.66f; 2.1^; .Of; .04f ; .OOf; 36.000|. 

Note. — Numbers expressed in both decimals and common fractions 
are called complex decimals. A simple decimal is expressed without the 
use of common fractions. 

To reduce such an expression as 3.444 to a mixed number pro- 
ceed thus: 

Or. .h„, 8.44, = 8M= 8;g= a|g j;_,?». 

9. Beduce the decimals of problem 8 to mixed numbers or 
common fractions. 

10. Make a rule for expressing any mixed decimal as a common 
fraction, or a mixed number. 

Principle I. — Any decimal may be expressed as a common 

fraction in its lowest terms or as a mixed nuinher^ hy dropping the 

decimal pointy writing the denominator^ and reducing the resulting 

common fraction to its lowest terms. 

§122. Bain and Snowfall (addition). 

Definition. — 1 in. of rainfall means a fall of 1 cu. in. of water on 
each square inch of surface of the ground. (Review §45, pp. 56-7). 

1. The following quantities of rain fell from week to week 
during May, 1902, in Chicago; find the total rainfall for the 
month: First week, 1.09 in.; second week, 1.11 in.; third week, 
.45 in., and fourth week, 2.43 inches. 
Solution.— 
CtoNVBNiBNT Form EXPLANATION.— It is convenient to write the 

1.09 m. addends in a column so that the units digits are all 
Ik ^"* ^^ ***® same vertical column. Then begin on the 
04Q •^* right and add as with whole numbers. In the sum 
2.43 in. the point should stand directly under the points in 
Ans, 5.08 in. the addends. 

Definition. — Finding the sum of decimal numbers is called addition 
of decimals. 



194 RATIOKAL GRAMMAR SCHOOL ARITHMETIC 

2. The following numbers denote the monthly rain or snow- 
fall for 1902; what was the total rain or snowfall for the year? 

.66 1.53 4.16 2.26 5.08 6.45 4.25 1.44 483 1.45 2.03 1.90 

3. Without rewriting the numbers, find the total yearly rain 
or snowfall for the years 1891-1901 from these recorded data: 

Jan. FSB. Mar. Apb. May Junb July Aug. Sbpt. Ocrr. Nov. Dec. Total 

1.99 1.95 2.13 2.14 2.09 2.42 2.47 4.52 .82 .36 3.83 1.32 

1.99 1.57 2.21 2.17 6.77 10.58 2.23 1.85 1.34 1.54 2.68 1.63 

2.08 2.44 1.69 416 1.93 8.59 3.08 .18 1.98 1.75 2 45 2.14 

1.55 2.13 2.66 2.65 3.35 1.96 .60 .60 8.28 .85 1.18 1.66 

2.15 1.60 1.32 36 1.99 1.79 2.43 6.49 1.89 .51 5.60 6.76 

1.12 3.48 1.26 2.79 416 2.82 3.61 3.52 6.70 1.36 2.16 .16 

4.53 2.22 3.56 2.23 .84 3.60 1.47 1.70 .84 .18 3.06 1.62 

3.54 2.59 460 .76 2.23 5.30 1.94 3.03 3.16 3.26 2.25 1.11 
.58 1.60 2.11 .14 4.35 2.71 6.66 .91 2.39 2.09 1.14 6.81 

1.21 3.52 1.58 1.02 3.59 2.06 4.64 4.24 1.56 1.35 3.30 .58 
1.15 2.05 3.38 .33 2.18 2.42 4.25 2.00 2.92 1.29 .85 1.70 

4. Foot and average the vertical columns and tell what the 
footings and averages mean. 

§123. Other Applications. 

1. A coal dealer received 8 carloads of coal of the following 
tonnages: 24.6, 28.785, 31.25, 24.95, 31.8, 25.125, 28, and 29.25. 
What was the total tonnage (number of tons)? 

2. A farm was divided by its owner iuto lots of the following 
acreages (number of acres) : 32.87^, 7.12^, 68.33^, 11.66|, 16.28t, 
and 21.13f . What was the total area of the farm? 

3. The following numbers represent in thousands of feet a 
lumber dealer's sales in 1 da. : 6.865, 24.245, 16.398. 12.28, 18.2, 
6.395, 24, and 18.967. What were the total sales for the day? 

4. A quantity of soil contained .125 lb. gravel; .268 lb. coarse 
sand; .175 lb. fine sand; .037^ lb. organic matter; .214f lb. clay, 
and .275 lb. water. What was the total weight of the soil? 

6. During 8 hr. a freight train made the following mileages: 
32.15, 28.375, 15.687, 20.2, 15.63, 17.5, 8.95, and 21.3. How 
far did the train run during the 8 hours? 

6. Following are the weights in grains of the U. S. coins: 
If-piece, 48; 5ji^-piece, 73.166f ; dime, 38.583^; quarter dollar, 



DECIMAL FRACTIONS 



196 



96.45; half dollar, 192.9; dollar, 412^; quarter eagle, 64J^; half 
eagle, 129; eagle, 258; double eagle, 516. Find the total weight 
of all, 

§124. Nature Study — (subtraction). 

1. 54 en. in. of soil in its natural state weighed 1.94 lb. 

After being thoroughly dried it weighed 1.459 lb. How much 

moisture passed off in drying? 

Explanation.— We liave seen that 1.94 may be 
written 1.940. For convenience write the numbers 
so that units digits stand in the same column. Be- 
ginning on the right subtract as though the numbers 
were whole numbers. In the result the point should 
stand directly under the points in the minuend and 
subtrahend. 



Solution.— 
Convenient Form 

1.940 lb. 
1.459 lb. 



Ans. .481 lb. 



DSFiNrnoN. — ^Finding the difference of decimal numbers is called 
subtraction of deeimah. 

2. After drying, the same soil occupied only 37.125 cu. in. 
How much did it shrink in bulk in drying? 

3. 100 green oak leaves weighed .22 lb. After thorough 
drying they weighed .087 lb. What was the weight of water 
contained in the green Iqaves? 

4. The organic matter in the leaves was then driven off by 
burning the dry leaves. The ash weighed .0053 lb. How much 
organic matter did the 100 leaves contain? 

5. The corresponding numbers for 100 green elm leaves were : 
Weight of green leaves, .132 lb. ; weight of dry leaves, .0345 lb. ; 
weight of ash, .0035 lb. Answer questions like 3 and 4 for these 
leaves. 

6. Answer similar questions for these leaves: 



Kind or Leaves 


Weight in Pounds 


Fresh, Gbeen 


Dbt 


Ash 


50 Poplar leaves 

35 Compound ash leaves. 


.132 

.099 


.043 
.044 


.0043 
.0026 



7. 1.135 lb. of dry beans, soaked for 24 hr., weighed 2.212 
lb. What was the amount of water taken up by the beans? 



19G 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



8. The dry beans occapied 34.875 en. in. and the soaKed 
beans 85.5 ca. in. How much did the beans increase in 
bulk? 

§125. Stature and Weight of Persons. 

The following table contains the average heights and weights of 
boys, girls, men, and women for the ages indicated by the numbers 
in the first column. Heights are given in feet and weights in 
pounds. 





Heights 




Growth in 
Height 


WmOHTS 




GROWTH ni 

Weight 


Age 
Yr. 






Difl. 






Dlff. 




















2 


Males 


Females 




Males 


Females 


Males 


Females 




Males Females 


2.60 


2.56 








25.01 


23.58 






. 


4 


8.04 


3.00 








31.38 


28.67 








6 


3.44 


3.38 








38.80 


35.29 








9 


4.00 


3.92 








49.95 


47.10 








11 


4.36 


426 








59.77 


56.57 








13 


472 


460 








75.81 


72.65 








15 


5.07 


492 








96.40 


89.04 








17 


5 36 


5.10 








116.56 


10443 








18 


5.44 


5.13 








127.59 


112.55 








20 


5.49 


5.16 








132.46 


115.30 








30 


5.52 


5.18 








140.38 


119.82 








40 


5.52 


5.18 








140.42 


121.81 








50 


5.49 


5.04 








139.96 


123.86 








60 


5.38 


497 








136.07 


119.76 








70 


5.32 


497 








131.27 


113.60 








80 


5.29 


494 








127.54 


108.80 








90 


5.29 


494 








127.64' 108.81 









1. Fill out on a separate slip the columns headed Difference 
(Diff.); the first by subtracting the height of the females from 
that of the males of the same age, and the second by subtracting 
the weights of females from those of males of the same age. Sab- 
tract without rewriting the numbers. 

2. There are 4 vacant columns headed Growth; two for growth 
in height and two for growth in weight. Fill out on a separate 
slip columns like the first of these from the column of heights of 
males by subtracting each number of this column from the one 



DECIMAL FRACTIONS 



197 



next below it. Tell what the difference means. Fill out a column 
like the second similarly from the column of heights of females. 

3. In a similar way fill out the last two columns from the 
columns of weights of males and weights of females. Tell what 
these differences mean. 

4. At what age are boys growing most rapidly in height? in 
weight? At what age do men begin to decrease in height? in 
weight? 

5. Answer questions similar to 4 for females. 

6. Compare your own height and weight with the numbers of 
this table, for your age. 

Note. — ^If your age is not in the table, it will be between two ages 
given there. Use the mean of the numbers for these two ages for your 
comparison.. 

7. The following table contains the height in feet of children 
of Manchester and of Stockport, (1) who are working in factories, 
and (2) who are not working in factories. Without rewriting the 
numbers, fill out the vacant columns of differences between the 
heights of the two classes for both boys and girls. What effect, 
if any, of such work can you detect (find out) on the growth of 
the boy or girl? 

Note. — When the boy or girl not working in factories is taller than 
the one forking in factories, mark the difference with a plus {-{-) sign 
before it. In the opposite case mark the difference with the minus ( - ) 
sign before it. 





Boys 




GTRT<8 




Agbs 






DIFT. 






Dirr. 




Working 
In Factories 


Not Working 




Working 
in Factories 


Not Working 






in Factories 




in Factories 




9 years.. 


4.009 


4.045 


8.996 


4.036 




10 - 


4.167 


4.219 




4.134 


4.114 




11 •• 


4.272 


4.252 




4.261 


4341 




12 •' 


4.446 


4.413 




4.475 


. 4.472 




13 •* ' 


4.537 


4.580 




4.636 


4.590 




14 " 


4.715 


4.725 




4.813 


4.852 




15 " 


4.971 


4.836 




4.875 


4.928 




16 " 


5.134 


5.266 




4.990 


5.003 




17 " 


5 228 


5.338 




5.039 


5.059 




18 " 


5.276 


5.825 




5.226 


5397 





198 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

§126. Pointing the Prodnet of Deeimals— (multiplication). 

ORAL WORK 

1. What is the relation between the following pairs of num- 
bers? 

(1) 25 and 2.5 (4) 1.28 and 12.8 (7) 2847 and 28.4? 

(2) 25 and .25 (5) 1.28 and 128 (8) 284.7 and 2.847 

(3) 75 and 7.5 (6) 47.8 and 4.78 (9) 28.47 and .2847 

2. The product 37 x 25 equals how many times the product 
37x2.5? 

3. The product 684 x 7.5 equals what part of the product 
684 X 75? 

4. What is the product 684 x 75? What, then, is the product 
684 X 7.5? What is the product 684 x .75? 684 x .075? 68.4 x.075? 
6.84 X. 075? .684 x.075? 

Definition.— By the number of decimal places of a number is meant 
the number of digits (zero included) on the right of the decimal point. 
Illustration. — In 2.005 there are 3 decimal place& 

WRITTEN WORK 

1. Find the product 4862 x 784, and from it write the follow- 
ing products : 

486.2 X 784; 48.62 x 784; 48.62 x 78.4; 4.862 x 7.84; 4862 x .784. 

2. How many decimal places are there in 486.2? in 48.62? 
4.862? 10.03? 3.0060? .0600? 20.0806? 

3. How many decimal places are there in each of the prod- 
ucts of problem 1? 

4. Compare the number of decimal places in each of the prod- 
ucts of problem 1 with the sum of the numbers of decimal places 
in both the multiplicand and the multiplier. What do you find? 

5. Make a rule for finding how many decimal places there 
must be in the product of two decimals. 

6. How, then, can you find where the decimal point belongs 
in the product of any two decimals? 



DECIMAL FRACTI0K8 190 

Principle II. — The number of decimal places in the product 
equals the sum of the numbers of decimal places in the factors, 

7. A field containing 38.75 A. yielded 23.9 bu. of wheat 

per acre ; what was the total yield? 

Solution. — Since each acre yielded 23.9 bu., 88.75 A. must yield 
88.75 X 23.9 bushels. 

CONYENUBirT FORM EXPLANATION. — 

^m ^«I5 239 X 38.75 = how many times 23.9 X 38.75? 

23.9 239 239X3875= ** *' 239x38.75? 

239 X 3875 = " " 23.9 X 38.75? 

What is irftto of 926,126? 
Does the rule of problem 5 hold true here? 

926.125 926125 
Ana. 926.125 bu. 

8. A steer weighing 16.22 cwt. sold at $8.85 per cwt. ; what 
price did he bring? 

9. A passenger train ran for 12.27 hr. at the rate of G5.75 
mi. per hour; how far did it run during the time? 

§127. Force Needed to Draw Loads on Eoad Wagon. 

1. To draw a load in a road wagon at a slow walk over hard, 
level country roads a horizontal pull of .075 of the total weight 
of the wagon and load is required. What horizontal pulls will be 
required to draw the following : 



1195 


1195 


1678 


1673 


1912 


1912 


717 


717 



Weight 
or Wagon 


Weight 
or Load 


Total 
Weight 


Horizontal 
Pull in L.b. 


(1) 1068 lb. 


2168 lb. 


.... 


.... 


(2) 969.5 


2408.3 


.... 


. • • • 


(3) 1580.75 . 


3675.6 


.... 


.... 


(4) 2368 


3890 


.... 


.... 



2. Over fresh earth .125 of the total load as a horizontal 
pull will draw it on a road wagon. Fill out the vacant columns, 
for these conditions : 



WlIOBT 
OF WAQON 


Weight 
OF Load 


Total 
Weight 


Horizontal 
Pull in Lb. 


(1) 980 lb. 


1260 lb. 


.... 


.... 


(2) 940.8 


768.78 


.... 


.... 


(3) 1164 


3890.85 


.... 


.... 


(4) 1675 


4060.75 


.... 





200 



RATIONAL GEAMMAB SCHOOL ARITHMETIC 



3. Over loose sand .258 of the total load will draw it. Find 
the forces needed to draw these loads : 



Wbight of 
Wagon 


WBIGHT OF 
LOAD 


Total 
Weight 


HOSIZONTAI. 
PULIi IN Lb. 


(1) 375 1b. 


686.9 lb. 


.... 


.... 


(2) 1860 


2586.38 


.... 


• . ■ • 


(3) 2800 


3869.25 


«... 


• ■ . . 



4. On good, broken stone pavement the pull is about .0285 of 
the total load; on wood pavement it is .019 of the total load; 
and on Macadam pavement it is .0333 of the load. Find the 
pull in pounds needed for each of these three kinds of pavement 
for the following: 



WmOHT OF 

Wagon 


Weight of 
Load 


Total 
Weight 


Horizontal 

PULL IN Lb. 


(1) 4060 lb. 


3795.65 lb. 


.... 


.... 


(2) 5190 


6340.86 


.... 


.... 


(3) 4960 


7640.65 


.... 


• . • • 



§128. DiviBion by an Integer. 

1. I paid $941.25 for 15 A: of land; what was the price per 
acre? 



Explanation. — Compare the steps in 
the work with decimals with the corre- 
sponding steps in the work with Integers. 

How do the decimal points stand 
through the problem? How does the point 
stand in the quotient? How may the 
dividend be found from the divisor and the 
Quotient? How, then, may you check 
division? 



Ana. $62.75. 

2. I paid $2823.75 for 45 A. of land; what was the price paid 
per acre? 



Convenient Form 


With Decimals 


With Integers 


62.75 


6275 


15)941.26 
90 


15)94125 
90 


41 
30 


30 


11.2 
10.5 


112 
105 


.75 
.76 


75 
75 



DECIMAL FRACTIONS 



201 



3. The 36 members of a society were assessed (made to pay) 
equally to meet a debt of $1341 against the society. What was 
the amount of the assessment against each member? 



Solution. — 
87.25 



36)1341.00 
108 

"iii 

252 

9.0 
7.2 

lio 

1.80 



Notice particularly how by writing zeros after the deci- 
mal point the quotient may be carried out in a decimal 
form. 

What effect does writing zeros after the point have on 
a number? 



Ans. $37.25. 



4. The number of states and territories and the total areas in 
square miles of the land surface of the principal geographic divisions 
of continental United States according to the Twelfth Census, are 
given in the following table. Find the average land surface of a 
state for each division and for the whole United States. Carry 
the division to three decimal places.* 



Division 


Number 


Land Subfacb 


Average 


North Atlantic 


9 
9 

12 
9 

11 


162,103 
168.620 
753.550 
610,215 
1,175,742 


. 


South Atlantic 


North Central 


South Central 


Western 


Continental United States 









6. The 10 loads given in pounds in column 2 of the following 
table required the number of pounds of force given in the third 
column, to draw them on a common road wagon over a good, level 
country road. For example, 1,400 lb. required 98 lb. to pull it; 



«If the next decimal place after the last one required is less than 5, write the given 
quotient as the required decimal. If it is greater than 5, add 1 to the last figure of 
tbe decimal required. This rule applies to all cases of division of decimals. 



202 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



1616 lb. required 112 lb., and so on. Divide each load by the 
number of pounds of force needed to draw it, and put the quotient 
to two decimal places in the column headed "Ratio." 



•BSIMENT 


LOAD 


Puu. 


1 


1400 


98 


2 


1616 


112 


3 


1825 


126 


4 


2236 


155 


5 


2440 


170 


6 


2650 


185 


7 


2863 


198 


8 


3072 


216 


9 


3281 


229 


10 


3662 


244 



Ratio 



§129. DiviBion by a Decimal. 

1. What is Vt ot 55? 

2. When no decimal point is written with a number where is 
it understood to be? 

3. Where is the decimal point in 55? 

4. How is a number changed by moving its decimal point 1 
place toward the left? 3 places toward the left? 1 place toward the 
right? 2 places toward the right? 

5. 55 equals how many times 5.5? .55? .055? 

6. If 55 + 11 = 5, to what is 5.5 ^ 11 equal? ,55 ^ 11? 

7. Name these quotients : 250 -s- 25 ; 25 -f- 25 ; 2.5 -i- 25 ; .25 + 25. 

8. Name these quotients: 250 -*- 2.5; 25 -*- 2.5; 25 -*- .25; 
250-^^.25. 

9. 2176 + 32 = 68; to what number is x equal in each of the 
following equations : 

(1) 217.6 ^32 = a:; (4) 217.6 + 3.2 = a?; (7) 2176 ^ 3.2 = a;; 

(2) 21.76-^32 = 2;; (5) 217.6 -^ .32 = a;; (8) 21.76+ .32 = a;; 

(3) 2.176 + 32 = a;; (6) 21.76 + 3.2 = a;; (9) 2.176 + .032 = a:? 

10. In each case of problem 9 compare the number of decimal 
places in the quotient with the number of decimal places in the 
dividend minus the number in the divisor. 



DECIMAL FK ACTIONS 



203 



11. 101,388-426 = 238; without dividing,write out the numbers 
to which X is equal in the following equations: 

(1) 10138.8-*-426=a;; (4) 101388 H-42.6=a;; (7) 101.3884- 42.6=2;; 

(2) 1013.88-^426=2;; (5) 101388 -^4.26=2;; (8) 10.1388-*- .426=2;; 

(3) 1.01388-^426=2;; (6) 1013. 88 -^4. 26= a;; (9) 1.01 388 4-. 0426 =2;. 

12. From the results of problems 10 and 11 make a rule for 
finding the number of decimal places in the quotient. 

Principle III. — The number of decimal places in the quotient 
equals the number of decimal places in the dividend minus the num- 
ber of decimal places in the divisor. 

§130. Problems. 

1. In 1891 the total imports of tea into 
U. S. were 82,395,924 lb. and of coffee 
611,041,459 lb. If the average cost of tea 
was 37^ per lb., and of coffee 18^, what 
was the total cost of both? 

2. On Dec. 6, 1901, a carload of 34 
Angus show cattle of weights given in table 
annexed sold at the prices per cwt. set 
beside the weights. What was the total 
weight of the carload? the average weight 
of the animals? 

3. What was the average price per 
hundredweight? 

4. For how much did the entire load 
sell? 

5. What was the average price per 
head which the owner received for the 
carload? 

6. The previous year on the same 
occasion, a carload of 26 prize-winning 
Angus cattle, averaging 1492 lb., sold at 
$15.50 per hundredweight. What aver- 
age price did the cattle bring the owner? 

7. How much did he receive for the 
carload, problem 6? 



No. 


Weight 


Prigb 


1 


1503 


$ 9.00 


2 


1622 


8.85 


3 


1606 


8.60 


4 


1524 


8.50 


5 


1524 


8.50 


6 


1504 


8.10 


7 


936 


8.70 


8 


1327 


6.85 


9 


1273 


8.05 


10 


1130 


8.75 


11 


1326 


7.65 


12 


1141 


8.50 


13 


1318 


8.75 


14 


1327 


8.70 


15 


1190 


7.70 


16 


1446 


7.85 


17 


1449 


7.85 


18 


1542 


7.70 


19 


980 


6.80 


20 


1460 


8.10 


21 


852 


8.30 


22 


1376 


8.30 


23 


1540 


25.00 


24 


1631 


9.30 


25 


1468 


8.65 


26 


1297 


8.50 


27 


1529 


8.20 


28 


1100 


7.60 


29 


1073 


7.60 


30 


1110 


8.10 


31 


1095 


8.15 


32 


1456 


8.75 


33 


1298 


8.00 


34 


2130 


10.75 



204 



BATIOKAL QRAMMAR SCHOOL ARITHMETIC 



8. A dairyman finds that during November one of his cows 
furnished this record : 





Morning Milkings 


Evening Miuungs 


Ist week 


47.2 lb. 

68.6 " 
58.8 *' 

62.7 ** 

17.8 *' 


40.4 lb. 
48.0 '• 
49.8 " 

47.3 - 

16.4 " 


2d " 

3d " 


4th " 


29th and 30th davs 





What was the total number of pounds of milk given by this 
cow during the month at the morning milkings? at the evening 
milkings? at both milkings? 

9. If milk was worth 6.25^ a quart (8.6 lb. per gallon), what 
was the milk of this one cow worth to the owner during November? 

§131. Batio of Circumference of Circle to Diameter. 

1. The distance around the rung of a chair was measured and 
found to be 3.625 in.; the diameter was found to be 1.153 in. 
Divide the distance around the rung by the diameter and find the 
quotient to 3 decimal places. 

Definition. — The distance around a circle is the circumference of the 
circle. 

2. The circumference of a circular rod, 1.875 in. in diameter, 
was measured and found to be 5.884 in. Find the ratio to 
3 decimal places, of the circumference to the diameter. 

3. Measure the diameters and the circumferences of any circles 
in your schoolroom and find to 3 decimal places the ratio of their 
circumferences to their diameters. If no circles are at hand use 
the measures of this table : 



Object 


CIRCUMPBRBNOB 


DIAMBTBR 


Ratio 


Ink bottle 


5.5 in. 

6.154 
26.538 
57.080 
71.458 
18.125 


1.75 in. 

1.953 

8.444 
18.062 
22.750 

5.675 




Tin box 


Globe of lamp 

Terrestrial globe 

Barrel-head 

Iron ring 


Average ...... 









DECIMAL FRACTIONS 



205 



4. Measure the circumferences and the diameters of the fol- 
lowing objects and find the ratios to three decimal places. If you 
are unable to make your own measures use those of the table: 



Object 


ClBCUMrERENCB 


Diameter 


Ratio 


Bicycle wheel 

Bicycle wheel 

Front carriage wheel 
Rear 

Locomotive driver . . 
Averasre 


81.177 

88.055 

151.189 

176.333 

174.762 


26.025 
' 28.012 
48.125 
56.125 
65.625 











5. Find the ratio of the circumferences to the diameters of the 
coins of problem 27, p. 108. 

6. It is proved in Geometry that the ratio of the circumference 
of any circle to its diameter is about 3^^, or more accurately 3.1416. 
If then, the diameter of a circle is known, how may the circum- 
ference be found without measurement? 

7. Find the average of all the values of the ratios found in prob- 
lems 1 to 4 and compare this last average with 3^; with 3.1416. 
What is the difference in each case? 

Note. — For most practical purposes this ratio, denoted by the Greek 
letter ir, and called pi, may be taken as 3k For greater accuracy use ir = 
3.1416. 

§132. Original Problems. 

Make and solve problems based on the following facts : 

1. 1 cu. ft. is about .8 of a bushel of small grain. A grain bin 
is 8' X 12' X 22'. 

2. 1 bn. of ear corn is about 2.25 cu. ft. A wagon box is 2.G7' 
X 2.85' X 9.6'. 

3. Well settled timothy hay runs about 355.25 cu. ft. to the ton. 
A hay shed is 24' x 40' and is filled with hay to a height of 18.5'. 

4. Loose timothy hay runs about 460 cu. ft. to the ton. A 
load of hay is 8' x 18' x 22.7'. 

5. Stove coal runs about 35.1 cu. ft. to the ton. A coal bin is 
6' X 12.5' X 15.35'. 

6. 1 perch of stone is 24.75 cu. ft. A stone wall is 2.75' x 
4.385' X 126.8'. 



206 



RATIOKAL GRAMMAR SCHOOL ARITHMETIC 



7. A man walks about 3.5 mi. per hour. It is 85 mi. from 
Chicago to Milwaukee. 

8. A horse trots about 7.5 mi. per hour. 

9. A horse runs about 18 mi. per hour for short distances. 

10. A steamboat runs 18 mi. per hour. 

11. A slow river flows 3 mi. per hour. 

12. A rapid river flows 7 mi. per hour. ' 

13. A crow flies 25 mi. per hour; a falcon, 75 mi.; a wild 
duck, 90 mi. ; a sparrow, 92 mi. ; and a hawk, 150 mi. per hour. 

14. A carrier pigeon flies 80 mi. per hour for long distances. 

15. Sound travels through air 1134 ft. per second; through 
water, 5000 ft. per second; and through iron or steel, 17,000 ft. 
per second. 

16. A rifle ball travels 1460 ft. per second at starting; and a 
20-lb. cannon ball, 16,000 ft. per second at starting. 

17. Light travels 186,600 mi. per second; electricity, 288,000 
mi. per second. The sun is 93,000,000 mi. from the earth ; the 
moon, 240,000 miles. 

18. One horse-power raises 33,000 lb. through a height of 1 ft. 
in 1 minute. 

19. The equatorial diameter of the earth is 7925.6 mi.; the 
polar diameter, 7899.1 miles. 

§133. Physical MeasurementB. 

The following table gives, for men, of ages from 18 to 26 years, 
the average lung capacity in cubic inches, the height in inches, 
and the weight in pounds. 



AGE 


Lung Capacity 


Height 


Weight 


18 


251.4 


68.2 


134.25 


19 


251.8 


68.2 


135.40 


20 


258.2 


68.1 


138.65 


21 


260.4 


68.1 


140.60 


22 


264.8 


68.2 


141.15 


23 


263.7 


68.1 


138.60 


24 


267.1 


68.2 


143.90 


25 


267.2 


68.2 


143.15 


26 


267.1 


68.9 


142.80 



DECIMAL FRACTIONS 



207 



1. Find the number of cubic inches of lung capacity per inch 
of height, for each age, by computing (finding) the ratio to 2 deci- 
mal places of each number of column 2 to the corresponding num- 
ber of column 3. Is this ratio the same for all ages? 

2. Similarly find the number of cubic inches, to 2 decimal 
places, of lung capacity per pound of weight, for each age. 

3. For each age find the number of pounds of weight per 
inch of height to 2 decimal places. Are these numbers the same 
for all ages? 

§134. Specific Gravity. 

The specific gravity of any solid or liquid substance is the 
ratio of its weight to the weight of an equal bulk of water. The 
weight of a cubic foot of water is G2.5 pounds. 

1. The following table contains the weight in pounds of 1 cu. 
ft. of the substances mentioned. Find to 3 decimal places the 
specific gravities of these substances : 



Metal 



Aluminum. 

Zinc 

Cast iron . . 

Tin 

Wr't iron. . 

Steel 

Brass 

Copper 

Silver 

Lead 

Gold 

Platinum . 






166.5 
436.5 
450.0 
458.3 
480.0 
490.0 
628.8 
552.0 
655.1 
709.4 
1200.9 
1847.0 



mo 



Wood 



Cork 

Spruce . . 
Pine (yellow) 

Cedar 

Pine (white) 

Walnut 

Maple 

Ash 

Beech 

Oak 

Ebony 

Lignum vitae 






(1)0 



15.00 
3L25 
84.60 
35.06 
28.00 
41.90 
46.88 
52.80 
53.25 
65.00 
76.00 
83.30 



Liquid 



Alcohol . . . 
Turpentine 
Petroleum 
Olive oil . . 
Linseed oil 
Sea water . 

Milk 

Acetic acid 
Muriatic acid 
Nitric acid. 
Sulphuric acid 
Mercury . 



o 

o . 

|2 



50.0 
54.4 
55.7 
57.0 
59.5 
64.1 
64.5 
66.5 
75.0 
95.0 
115.1 
880.0 



mo 



2. The specific gravity of any gas or vapor, is the ratio of its 
weight to the weight of an equal volume of air, the gas or vapor 
and the air being at the same temperature and under the same 



208 



RATIONAL GRAMMAR 8CH00L ARITHMETIC 



pressure. Find to 3 decimal places the specific gravities of the 
gases and vapors in the following table : 



Gas 


Weight, in Pounds, 
or 1 Cubic Foot 


Specific Gravity 


Hvdrofiren 


.00559 
.00727 
.00815 
.03790 
.07810 
.07860 
.08073 
.08925 
.12344 
.19700 




Smoke (wood) 


Smoke fsof t coal) 


Steam at 212° F 


Carbonic oxide 


Nitro&ren 


Air 


Oxygen 


Carbonic acid 


Chlorine 





3. The following familiar substances may be compared with 
water as to weight. Find their specific gravities to 2 decimal 
places : 



Substance 


Weight, in Pounds, 
OF 1 Cubic Foot 


Specific Gravity 


Glass (average) 


175.8 
174.5 
169.2 
166.4 
158.2 
133.4 
124.5 
121.8 
118.3 
113.9 




Chalk 


Marble 


Granite 


Stone (common) 


Salt " 


Soil " 


Clay 


Brick 


Sand 





§135. To Eeduce a Common Fraction to a Decimal. 

1. Express $f decimally. 
Solution 



.375 

8)3.000 
2.4 

"jftO 
.66 

.040 
.040 



Explanation. — Annex zeros to the right of the 
decimal point after the numerator, and then divide by 
the denominator. 

^Ins. $i=$.875 



DECIMAL FBACTI0N8 209 

2. Express the following common fractions decimally (to 3 
decimal places) : 

il il i> f; fj TIT 5 /?> H« 

3. Express the following mixed numbers decimally (to 3 
places) : 

H; 3i; 2i; 6f; 13^^; 18 Jf 

4. Express the following decimally to 5 decimal places: 

^5 tVj tt> ifV; H> A- 

Such decimals as these fractions give rise to, that do not ter- 
minate, are called non'terminating decimals. 

5. Express the following numbers decimally to 6 decimal 
places : 

i;l; \\ f; i; I; i; i*t; A; tV; 4?; H; If 

DEFmrriON. — Such non-terminating decimals as these that repeat the 
same digit or group of digits indefinitely, are called repetends, or circula- 
ting deciTnals, or circulates. 

6. Express the values of these numbers by the use of integers 
and decimals only : 

1.3i; 16.87i; 5.281; 17.9|; 20.0|; l.OOf; .OOJ ; .OJ; .000,^; 
30.060f, 

7. Express the following fractions decimally to 4 places: 

ft, t^, tNf, TT> ^^, ^y 7 y, 12.17' 12f 

8. A man sold 37i A. of his farm of 79 J A. ; how many 
hundredths of his farm did he sell? 

9. A man owned a piece of city land 450' square. A strip 
37 J' wide was cut from each of its four sides for streets. How 
many hundredths of the square were cut away? 

10. How many hundredths of the area of a page of this book 
are in the margins? (Measure to the nearest 16th of an inch). 

11. How many hundredths of the area of the surface of your 
desk is the area of the surface of your book? 

12. How many hundredths of the area of the surface of the 
floor is the area of the surface of your desk? 



210 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

B. Aroa of a Circle. 



Draw a circle and diameter AB (see Fig. 132). With an 
opening of the compasses equal to the radius of the circle and with 




Figure 132 

A as center draw short arcs at D and G^ also, with same radius 
and with B as center, draw short arcs at E and F. Draw 
radii OD^ OE, OF, and OQ. 

Bisect angle AOD, as in Problem II, p. 176, and draw 08, 

With an opening of the compasses equal to the distance 
between D and H and with center D draw an arc at J\ with 
center B draw arcs at K and L ; also, with center O draw arcs at 
if and N. Draw radii OJ, OK, OL, OM, and ON. 

Cut the circle into the twelve equal sectors thus formed and 
place these sectors as shown in the second part of the figure. 
How long is the base AB oi the approximate parallelogram thus 
formed? How wide is the approximate parallelogram? 

If each of the twelve sectors were split into halves and the 
resulting twenty-four sectors were fitted together as are the twelve 
sectors, would the wavy base line become more nearly straight? 
How long and how wide would the new approximate parallelogram 
be? What would be its area? 

Having the circumference and the radius of a circle how can 
you find the area of the circle? 

Having the radius of any circle how can you find the area of 
the circle (See §131)? 

PROBLEMS 

In problems 1 to 3, inclusive, use «• = 3if. Let r denote the 
length of the radius of a circle, let c denote its circumference, 
and Ay its area. 

1. Find the areas, -4, of these circles 

(1) r = 12.5', c = 78.54'; (3) r = 20", c = 125.664"; 

(2) r = 6.25', c = 39.27'; (4) r = 60", c = 377.143". 



DECIMAL FRACTIONS 211 

2. The diameter of a drum-head is 2.5' and the circumference 
18 7.854\ how many square inches of skin are in the two heads? 

3. The diameter of a circular window is 18.5" and its circum- 
ference is 58.12", what is its area? 

In the following problems v is taken as 3.1416. Results 
should be correct to 4 decimal places. 

4. The wind is blowing squarely against a circular signboard, 
whose diameter is 32.75 ft., with a pressure of 25.5 lb. to the 
square foot. Find the total wind pressure against the board. 

5. The circumference of a cylindrical chair rung is 5.5"; find 

the diameter and area of the right section of the rung. 

Note. — A right section is the section that would be made by sawing 
the rung square across. 

6. Steam passes from the boiler of an engine through the 

^ passage, P, into the cylinder, and pushes 

jy^ vk against the circular piston, C^ with a pres- 

Y^\ — Ir , "7*^ sure of 65 lb. to the square inch. If the 

diameter of the circular head, C, is 12.5", 

what is the total pressure against the left side 
FiouiiE 133 ^f ^1^3 pjg^Q^p 

7. Answer similar qnestions for pistons having these diam- 
eters with steam pressures per square inch as indicated: 

DIAMBTEB OF PRESSURE PER 

PISTON Square Inch 

(1) 10.85" 86 

(2) 22.35" 120 

(3) 14.85" 180 

(4) 16.45" 175 

(5) 20.75" 125 

(6) 15.85" 160 

(7) 21.35" 205 

8. Find the area, ^, of a circle whose radius is r ft. long and 
whose circumference is c ft. long. 

9. Find the circumference, c, of a circle whose radius is 
r rods long. Find the area. 

10. A cow is tied to the corner of a corn crib, 18' x 18', with 
a rope 18' long. Over how many square feet can she graze? 




212 RATIONAL GRAMMAR SCHOOL ARITHMETIC 



§137. Exercises for Practice. 










1. Add the following : 










[I) 37.8 


(2) 248.001 


(3) .96 


(4) 7.68 


(5) 


862.01 


(6) 328. 


8.65 


16.25 


3.004 


86.043 




2.8106 


.032 


12.08 


9.018 


10. 


190.608 




843.7 


7.001} 


.75 


24. 


7. 


9. 




1200. 


29.080i 


16. 


3. 


.206 


.9 




.16 


109. 


14.01 


.03 


200. 


8.3 




9. 


9000. 


.6 


.4 


860.03 


60.5 




.0068 


.08 


10. 


17.003 


.2 


3.7 




.0309 


.0007 


2. Subtract the following : 










(1) 84.96 


(2) 98.01 


(3) 100.01 


(4) 106.038 


(5) 67.083 


(6) 800.00 


6.38 


1.97 


87.08 


69.879 


29.187 


98.68 



(7)810.02 (8)3.786(9)26. (10)3. (11)68. (12)1.0101 
86.983 .989 .748 2.983 9.8789 .9008 

3. Find the following products: 

(1) 89.3 X 42.1 (7) 8.08 x .008 (13) 31.416 x 2.518 

(2) 68.01 X 9.82 (8) 16.8 x .072 (14) 314.16 x 251.8 

(3) 86.3 X 8.761 (9) 9.001 x 1.801 (15) 785.4 x 2.781 

(4) 100.001 X 86.8 (10) 1.68 x 7.2 (16) .7854 x .2781 

(5) 76.9 X .93 (11) 168 x .072 (17) 3 1416 x 16.84 

(6) 7.69 X .093 (12) 3.1416 x 25.18 (18) 314.16 x .1684 

4. Find the following quotients : 

(1) 120.716 -^ 26.82 (5) 109.624 -^ 3.86 (9) 443.52 ^ .96 

(2) 123.1498 -^ 26.82 (6) 548.12 ^ 14.2 (10) 14.784 -^ .032 

(3) 276.496 -^ 3.142 (7) 1774.08 -*- .384 (11) .3696 ^ .0008 

(4) 268.928 ^ 8.4 (8) 8.8704 -^ 192 (12) 9.24 -^ .00004 

5. Find the following quotients to 3 decimal places: 
(1)8^.3 (7)2^3 (13)4-^.011 (19)1.74^9.09 

(2) 031 -^ .072 (8) 4 -*- 9 (14) 2.5 ^ .033 (20) .77 -*- 1.21 

(3) 9 ^ .007 (9) 44 -^ .09 (15) 260 -^ 3.3 (21) 1.111 4..099 

(4) 11 -^ .0009 (10) 13 H- 3 (16) 2.6 -^ 33 (22) 60.6 -^ 1.818 

(5) 12 H- 36 (11) 7.7 ^ 9 (17) 11.5 ^ 3.33 (23) .625 ^ 3.36 
(C) 1 -^ 3 (12) 123 ^ 9.99 (18) 10.4 ^ .909 (24) .1 -*- .09 



COMlPOUND DENOMINATE NUMBERS 213 

COMPOUND DENOMINATE NTJMBEBS 
§138. Definitions. 

A denominate number is a number whose unit is concrete; 
as,- 13 mi., 8 hr., 40 A., $125, etc. 

A concrete unit is a unit having a specific name; as, 1 yd., 
1 lb., $1, 1 hat, 1 horse, etc. 

A compound denominate number is a number expressed in two or 
more units of the same kind; as, 11 hr. 25 min. 15 sec; 12 gal. 
■J qt. 1 pt. 3 gills. 

TABLES OF MEASURES 

Note. — Read carefully the tables that follow and fix them in mind by 
solving the problems beginning on page 219. 

§139. Measures of Value. 

The standards of value of the United States and of some of 
the European countries are here given with both their rough and 
their accurate equivalents in U. S. money. 

Rough Accurate 
Country Standard Symbol Equivalen't Equivalent 

United States Dollar $ §1. $1. 

Great Britain Pound (Sterling) £ $5.00 $4.8665 

Germany Mark (Reichsmark) M. $ .25 $ .2385 

France Franc F. $ .20 $ .193 

Russia Ruble R. $ .75 | .772 

Austria-Hungary Crown, or Filler C. or F. $ .20 $ .203 

Italy Lira L. $ .20 $ .193 

Ta^le of U. S. Money 
10 mills (m.) = 1 cent (ct. or ^) 
10 cents = 1 dime (d.) 

10 dimes = 1 dollar 
10 dollars = 1 eagle 

5 dollars = | eagle 

2i dollars = J eagle 
20 dollars = 1 double eagle 

The coins of the United States are bronze, nickel, silver, and 
gold. 

Table of English Money 
4 farthings (far.) = 1 penny (d.) 
12 pence = 1 shilling (s.) 

20 shillings = 1 pound (£) 

21 shillings = 1 guinea 



214 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

Table of Monetary Units of Other Nations 
Germany, 1 mark (M.) = 100 pfennige (pf.) 

France, 1 franc (fr.) = 100 centimes (c.) 

Russia, 1 ruble (r.) = 100 copecks (c.) 

Austria-Hungary, 1 crown, or fiHer = 100 heller 
Italy, 1 lira =100 centessimi 

§140. Measures of Weight. 

Three systems of weight units are used in the United States, 
viz. : troy weight, avoirdupois weight, and apothecaries' weight. 

Troy weight is used in weighing gold, silver, and jewels. 
Avoirdupois weight is used for weighing all ordinary articles, and 
apothecaries' weight is used by druggists in mixing medicines. 

The standard of weight in the U. S. is the troy pound. The 
grain is the same in all three systems; the pound the same in tro} 
and in apothecaries' weight and different in avoirdupois. 



Table of Troy Weight Table of Equivalents 

24 grains (gr.) = 1 pennyweight (dwt) 5760 gr. \ 
20 penny weights = 1 ounce (oz.) 240 dwt. >• = 1 lb. 

12 ounces = 1 pound (lb.) 12 oz. ) 

Table of Avoirdupois Weight 
7000 grains (gr.) = 1 pound (lb.) 
16 ounces = 1 pound (lb.) 

100 pounds = 1 hundredweight (cwt.) 

2000 pounds = 1 ton (T.) 

2240 pounds = 1 long ton (L.T.) 

Table of Apothecaries' Weight 
20 grains (gr.) = 1 scruple (3) 

3 scruples = 1 dram (3) 

8 drams = 1 ounce (1) 

12 ounces = 1 pound (lb.) 

§141. Measures of Length, or Distance (Linear Measure). 

The standard unit for measures of length, or distance, is the 
yard. 

Table of Common Linear Measure Table of Equivalents 

12 inches (in.) = 1 foot (ft.) 63360 in. ^ 

3 feet =1 yard (yd.) 5280 ft. [ __ 

^ yards, or 161 ft. = 1 rod (rd.) 1760 yd. [ " ^ °^^ 

320 rods =1 mile (mi.) 320 rd. 



COMPOUND DENOMINATE NUMBERS 215 

Table op Surveyors' Linear Measure 

7.92 inches = llink (li.) 

100 links == 1 chain (ch.) 

80 chains = 1 mile (rai.) 

1^142. Measures of Surface. 

The unit upon which surface measure is based is the square 
yard^ which is a square each of whose sides equals 1 yard. 

Table op Common Surpace Measure 

144 square inches (sq. in.) = 1 square foot (sq. ft.) 
9 square feet = 1 square yard (sq. yd. ) 

30 J square yards = 1 square rod (sq. rd.) 

160 square rods = 1 acre (A.) 

Table op Surveyors' Surface Measure 

625 square links (sq. li. ) = 1 square rod (sq. rd. ) 
16 square rods = 1 square chain (sq. ch.) 

10 square chains = 1 acre (A.) 

640 acres (a section) = 1 square mile (sq. mi.) 
86 square miles = 1 township (Tp. ) 

Table of Equivalents 
3686400 sq. rd. j 
280400 sq. ch. [ = 1 Tp. 
36 sq. mi. ) 

§143. Measures of Volume. 

Table op Cubic Measure 
1728 cubic inches (cu. in.) = 1 cubic foot (ou. ft.) 
27 cubic feet = 1 cubic yard (cu. yd.) 

Table op Equivalents 

46656 cu. in. ) , , 

27c«.ft. {=!««• yd. 

A perch of stone is a square-cornered mass, 1' x 1|' x 16^, 
or 24i cubic feet. 

Fire wood is measured by the cord. A cord of wood is a 
straight pile, 4' x 4' x 8', or 128 cubic feet. A cord foot is a straight 
pile of wood, 4' x 4' x 1'. How many cubic feet are there in a 
cord foot? 



21G 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



§144. Measures of Capacity* 

Liquid measure is used in measuring liquids, the capacity of 
cisterns, tanks, etc. 



Table op Liquid Measure 
4 gills (gi.) = 1 pint (pt.) 
2 pints = 1 quart (qt. ) 
4 quarts = 1 gallon (gal. } 

Note. — A liquid gallon contains 231 cubic inches. 



Table op Equivalents 
32 gi. J , 
8 pt. [ = 1 gal. 
4qt. ) 



Dry measure is used in measuring grain, fruit, vegetables, etc. 
The standard of dry measure is the Winchester bushel^ which is 
a cylinder 18^ in. in diameter and 8 in. deep, containing 2150.42 
cubic inches. 



Table^of Equivalents 
64 pt. 
32 qt. S- = 1 bu. 



Table of Dry Measure 
2 pints (pt. ) =7= 1 quart (qt.) 
8 quarts = 1 peck (pk.) 
4 pecks = 1 bushel (bu. ) 

Note. — A dry gallon = 4 qt. or \ pk., or 268.8 cubic inches. 
avoirdupois pounds in a bushel prescribed by various states 



D4pt. \ 

32 qt. [ =\ 
4pk. ) 



COMMQUmZS, 



BBFlef .»....>' 

BeaD^ < 

Blue Grass Seed . 

Buckv'hetLt ,«i 

Castor Ubuqs — ^< 
CJcver Seed 

CorD on the Cob . , 
Cora, Sbelled.., ,. 

Cartimeai 

Dried Apples.,,.. 

Dried Pouches 

Flax Seed 

Bcmp Set^d .., 

Miilbt ..*„.. 

Oftts 

Onlana. ........... 

I>«i4B 

Potatoes .*.- 

Rye . , , 

Sweet Potatoes .. 
TlttitJtliy Seed .„. 

iTurcipB 

Wheat , . . „ , 



50 



EiO 



as: 



54 



50 48 

44 
nO 
32 
57 



45 



^ < 



52 50 

4a 

60 

m 

70 
5G 



.3 



47 4^' 
60W 
14 14 



fiO 



56 



46 
WflO 

70' 70 

50,50 

24 24 
33 39 
56 56 

44 44 

3S33 

57,67 
flO 
60,60 61) 

?fi '56^56 M' 
46 50"55i.J 

45 45 1 45' 

t^ m\ 
UU6U60 



m 



m 



4& 



4H 



6U 



|GU. 



62 



m 



5^ 



60100 



3S 



m 



2E24 
33 
55 
44 



tfy 



^ 



m 



m 



m- 



5fi 



47 



GO .. 
33 SO 
56 

60 «0 



56 



50.. 
45 



mm 






4S i% 4e 

^At«0 60 

50,42,4U 



,Jeo 

070 
56,56 

441 
50 



60 



fi6;57 
60 
00, 6D 
6C,5e 
50! 55 
4&i4& 
5& 
60.60j6Q 



48' 4e 
mm 
u 



5e 



6« 



45 41] 

@oe 



60t&7 



COMPOUND DENOMINATE NUMBERS 217 

§145. Measures of Time. 

The standard unit for measuring time is the mean solar day. 
The mean solar day is the average time interval from the instant 
when the sun crosses the meridian of a place (noon) to the next 
instant of crossing the meridian (the next noon). 

Table of Time Measure 
60 seconds (sec.) = 1 minute (min.) 
60 minutes = 1 hour (hr.) 

24 hours = 1 day (da. ) 

7 days = 1 week (wk.) 

30 days = 1 calendar month (mo.) 

12 calendar months = 1 calendar year (yr.) 

365 days = 1 common year 

366 days = 1 leap year 
100 years = 1 century 

Note. --One mean solar year = 365 da. 5 hr. 48 min. 46 sec. = 365} da. 
(nearly). 

Table op Equivalents 



31536000 sec. ^ 

525600 min. 
8760 hr. 
365 da. 
52 wk. 1 da 
12 mo. 



>. = 1 common yr. 



According to the Julian calendar^ adopted by Julius Caesar 
(whence the name), every year whose date number (as 1896, 1904) 
is divisible by 4 must contain 366 da. and all other years, contain 
365 da. Years containing 366 da. are called leap years; those 
containing 365 da. are called common years. 

According to the Gregorian calendar^ which is now used by 
nearly all civilized nations, every year whose date number is 
divisible by 4 is a leap year, unless the date number ends in 
two zeros (as 1600,- 1900), in which case the date number must 
be divisible by 400 to be a leap year. 

The extra day of the leap year is added to February, giving 
this month 29 da. in leap years. 

§146. Measurement by Counting. 

Table for Counting Table for Measuring Paper 
12 things = 1 dozen (doz.) 24 sheets = 1 quire 

20 things = 1 score 20 quires = 1 ream 

12 dozei;! = 1 gross 2 reams = 1 bundle 

12 gross = 1 great gross 5 bundles = 1 bale 



218 RATIONAL GBAMMAR SCHOOL ARITHMETIC 

The operations to be performed in compound denominate 
numbers are the following: 

(1) To change from 'higher to lower units,, or denominations; 

(2) To change from lower to higher units, or denominations; 

(3) To add compound denominate numbers; 

(4) To subtract such numbers; 

(5) To multiply such numbers; 

(6) To divide such numbers. 

These processes will now be illustrated in order. 
1. Express 16 bu. 3 pk. 3 qt. 1 pt. as pints. 
Convenient Form 

16 bu. 3 pk. 3 qt. 1 pt. 
4 = No. pk. in 1 bu. 



64 pk. = No. pk. in 16 bu. 
3pk. 



67 pk. = No. pk. in 16 bu. 3pk. 
8 = No. qt. in 1 pk. 



536 qt. = No. qt. in 67 pk. 
3qt. 

539 qt. = No. qt. in 67 pk. 3 qt 
2 = No. pt. in 1 qt. 

1078 pt. = No. pt. in 539 qt. 
1 pt. 

Am. 1079 pt. = No. pt. in 16 bu. 3 pk. 3 qt. 1 pt. 

2. Express (1) 1079 pt. in higher units; (2) 2 pk. 3 qt. 1 pt. 

in bushels. 

Convenient Form 

(1) 
2 ) 1079 pt. 

8 ) 539 qt. + 1 pt. remaining 

4 ) 67 pk. + 3 qt. remaining 

16 bu. + 3 pk. remaining 

Ans, 16 bu. 3 pk. 3 qt. 1 pt. 

(3) 

8 qt. 1 pt. = 3.5 qt. = ^ pk. = .4875 pk. 

2 pk. 3 qt. 1 pt. = 2.4375 pk. = ^^^ bu. = .609375 bu. Am. 



COMPOUND DENOMINATE NUMBERS 219 

3. The three sides of a triangular grass plot were: 8 yd. 2 ft. 

10 in. ; 12 yd. 1 ft. 9 in. ; and 9 yd. 2 ft. 7 in. ; how far is it 

around the plot? 

Explanation.— First, adding the inches, 
Convenient Form ^^ ^^^^^^ ^g j^ = 2 ft. 2 in. Write the 2 in. 

8 yd. 2 ft. 10 in. in 'inches" column, and add the 2 ft. to the 
12 1 9 numbers in the '"feet" column, giving 7 ft. 

9 2 7 = 2 yd. 1 ft. Write 1 ft. in '*feet" column 

Ans 31 vd 1 ft 2 in *^^ *^^ ? ^^' ^ ^^® numbers in **yards" col- 
^ ' ' umn, giving 31 yd. 

4. From a vessel containing 25 gal. 3 qt. 1 pt. 2 gi. of oil, 

8 gal. 3 qt. 1 pt. 3 gi. were drawn out; how much oil remained 

in the vessel? 

Explanation. --3 gi. can not be taken 

from 2 gi. But the 1 pt. of the minuend 

Convenient Form equals 4 gi. and this added to 2 gi. gives 

6 gi. 6 gi. — 3 gi. = 3 gi. 1 pt. from pt. 

25 gal. 2 qt 1 pt. 2 gi. can not be taken. But 1 qt. is taken from 

8 gal. 3 qt. 1 pt. 3 gi. 2qt., and changed to pints, giving 2 pt. 

Ans. 16 gal. 2 qt. 1 pt. 3 gi. Finally, 24 gal. — 8 gal. = 16 gal. Prove 

the work by adding the remainder to the 
subtrahend and comparing the result with 
the minuend. 

5. A man built an average of 45 ft. 8 in. of fence a day for 
16 da. ; how much fence did he build in 16 days? 

Convenient Form 

45ft. Sin. 

16 Explanation.— 
r- 8 in. X 16 = 128 in. = 10 ft. 8 in. 

10 ft. 8 m. 45 ft. X 16 = 720 ft. 

720 



^ns.730ft. Sin. 

6. An iron rod, 4 yd. 2 ft. 8 in. long, was cut into 5 equal 
pieces ; how long was each piece? 

Convenient Form ^ . , ., « i.^ ^ .^ i..^ . ^ -^ 

Explanation. —4 yd. =12 ft. 12 ft. + 2 ft. 
5 ) 4 yd. 2 ft. 8 m. = 14 ft. 14 ft. -*- 5 = 2, with a remainder of 
Ans. 2 ft. 114 in. ^ ^^' ^ f^- = ^^ i°- 48 in. + 8 in. = 56 in. 

§147. ExerciseB on Denominate Number Tables. 

standards of value 
1. IIow many cents are there in 23 dimes? in $23.00? in 23 



220 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

2. How many dollars are there in 248 cents? in 248 dimes? in 
248 eagles? 

3. How many half eagles are there in $245.00? in 240 dimes? 

4. About what is the value of S50.00 in pounds sterling (Eng- 
lish money)? in Marks? in Francs? in Rubles? in Crowns? in Lira? 

5. What are the more accurate equivalents of $100.00 in these 
same units? 

6. I owe a debt of 32 M. 50 Pf . (pfennig = y^^ of a Mark). 
How much U. S. money will it take to pay this debt if I have to 
pay 24.5 cents for a Mark? 

7. If I have to pay $4.94 for a pound sterling, how much U. S. 
money will it take to pay a debt in London of £65 8s.? 

WEIGHT 

1. How many grains are there in 

(1) 12 dwt. ; (3) Si oz. ; (5) 9 oz. 12 dwt. 

(2) 3 oz. ; (4) 8 oz. 10 dwt. ; (6) 4 lb. 8 oz.? 

2. A carat (of weight) usually means 32 troy grains. How 
many carats are there in a diamond weighing 13 dwt. 8 gr.? 

3. How many carats weight are there in a diamond weighing 2 
troy ounces? 

4. The avoirdupois table is used for weighing iron, coal, and 
all coarser (less costly) articles. How many grains heavier is a 
pound of coal than a pound of silver? How many troy ounces ? 

5. Express 8 lb. (av.) as ounces. Express 8 lb. (troy) as 
ounces. 

6. How many ounces are there in 24 lb. of iron? in 48 lb. 6 
oz. of iron ? 

7. How many ounces are there in 3 cwt.? in 1 cwt. 40 lb.? in 
5 cwt. 8 oz.? 

'8. Express as avoirdupois ounces the following avoirdupois 
weights : 

(1) 3 lb. 8 oz. ; (3) 86 lb. 15 oz. ; (5) 8 cwt. 13 lb. ; 

(2) 18 lb. 12 oz. ; (4) 1 cwt. 8 lb. ; (6) 9^ cwt. 
9. Express as tons the following: 

(1) 40 cwt.; (3) 16000 1b. (5) 12500 1b.; 

(2) 30 cwt. ; (4) 9000 lb. (6) 13500 lb. 



COMPOUND DENOMINATE NUMBERS 221 

10. Express the following as long tons : 

(1) 6720 lb. ; (3) 112 cwt. ; (5) 56 tons; 

(2) 19040 lb. ; (4) 952 cwt. ; (6) 39 cwt. 20 lb. 

11. How many pounds are there in ^ of a ton of coal? in ^ of 
a long ton of pig iron? 

LINEAR MEASURE 

1. How many inches make a rod? in 1 dhain? 
^ 2. How many feet are there in 1 chain? 

3. Express 3 rd. 2J yd. in inches. 2 rd. 1 yd. 2 ft. in inches. 

4. Express 6408 inches in rods and yards. 

5. A toy railroad track is 10 yd. long. How many feet of 
track are needed for it? 

6. A 12-rod fence is made of 10-inch boards set vertically with 
no cracks between them. How many boards are needed? 

7. The four sides of a garden are 10 rods, 1020 ft., 162^ ft., ■ 
and 1019 ft. 4 in. How many 10-inch boards will be needed to 
enclose it with a fence like that described in problem 6? 

8. The tire of a bicycle is 88 inches long. How many times 
will the wheel turn in going 40 rods? 

9. The tire of a wagon wheel is 15 ft. 9 in. long. How many 
turns will the wheel make in going 5 miles? 

SURFACE MEASURE 

1. Express as square inches : 

(1) 3 sq. ft. ; (4) 2 sq. yd. 3 sq. ft. ; (7) 3^ sq. yd. ; 

(2) 3 sq. ft. 8 8q. in. ; (5) 6 sq. yd. 48 sq. in. ; (8) 45^ sq. ft. ; 

(3) 1 sq. yd. (0) 45 sq. ft. 108 sq. in. ; (9) 1 sq. rd. 
2 Express: 

(1) 121 sq. yd. as sq. rd. ; (4) 80 s^. rd. as A. ; 

(2) 640 sq. rd. as A. ; (5) 1 sq. rd. as sq. ft. ; 

(3) 40 A. as sq. rd. ; (6) 1 A. as sq. yd. 

3. Express: 

(1) 10 sq. ch. as A. ; (4) 1 Tp. as A. ; 

(2) 20 A. as sq. ch. ; (5) 1 A. as sq. rd. ; 

(3) 1 sq. mi. as sq. cb. ; (6) A. section as sq. rd. 

4. How wide must a rectangle 80 rd. long be to contain 1 A.? 
5 A.? 20 A.? 80 A.? 



^22 flAttOHAL ORAMMAR SCHOOL ARttHMETlO 

5. How wide must a rectangle 16 rd. long be to contain 1 A? 
How many sq. rds. more than 1 A. are there in a 14-rd. sqnare? 

G. How many square yards more than 1 A. are there in a 70- 
yd. square? 

7. How many square yards less than 1 A. are there in a 69-yd. 
square? 

8. How many square feet are there in 1 A.? 

9. How many square yards more than 1 A. are there in a 21?- 
ft. square? 

CAPACITY 

1. Express: 

(1) 2 qt. 1 pt. as pt. ; (4) 2 gal. as pt. ; 

(2) 1 qt. i pt. as gi. ; (5) 12 gal. as qt. ; 

(3) 3 qt. 1 pt. 2 gi. as gi. ; (6) 11 gal. 3 qt. as pt. 

2. Express: 

(1) 24 pt. as qt. (4) G4 gi. as gal. 

(2) 48 gi. as qt. (5) 17 pt. as qt. and pt. 

(3) 76 pt. as gal. (6) 84 pt. as gal. and qt. 

3. There are 231 cu. in. in a liquid gallon. How many cubic 
inches make a quart? a pint? 

4. There are 268.8 cu. in. in a dry gallon. How many more 
cubic inches are there in a dry than in a liquid quart? 

5. Express: 

(1) 3 bu. as pk. ; (4) 4 bu. as dry gallons; 

(2) 2i bu. as pk. ; (5) 18 dry gallons as bu. ; 

(3) 2i bu. as pk. ; (6) 20 pk. as pt. 

6. What is the difference between the weight of a quart of 
barley in California and in Illinois? 

7. How many bushels are there in a long ton (2240 lb.) of 
anthracite in Illinois? in a short ton? 

8. My coal bin is 7'xl2'xl2'. How many bushels of anthracite 
will it hold? how many tons? (Count 1^ cu. ft. per bushel.) 

9. About what part of a Winchester bushel is 1 cubic foot? 

10. Express 3.5 bu. as pints; 160 pt. as bushels. 

11. A grocer buys beans at $1.80 per bu. and sells them at 9^ 
a qt. Compare the buying and selling prices per bushel ; per quart. 



COMPOUND DENOMINATE NUMBERS 



223 



12. Find the error in the adage *'A pint's a pound'' if there 
are 7^ gal. in a cubic foot of water weighing 1000 ounces. 

TIME 

1. Express: 

(1) 1 hr. as. sec. ; (4) 30 sec. as hr. ; (7) 1209600 sec. as wk. ; 

(2) 1 da. as sec. ; (5) 7200 sec. as hr. ; (8) 1000000 sec. as wk. 

da. and hr. 

(3) 1 wk. as sec; (6) 604800 sec. as da. ; (9) 1 wk. as minutes. 

2. How many weeks in a common year? July 4 falls on Tues- 
day in 1905. On what day of the week will July 4, 1906, fall? 
Why? 

3. How many months old are you this month? 

4. How many days old are you to-day? How many hours in 
this many days? 

5. A clerk earned $15 a week for 28 calendar months. How 
much did he earn in all? 

6. A clerk saved $6 a week for 7 yr. How much did he save 
in all? 

7. Illinois was admitted as a state in December, 1818. How 
many years and months old is the state of Illinois this month? • 

8. How many years and months old is your native state. (See 
Table, page 39.) 

9. On the Saturdays of April, 1905, the sun rose and set at 
Chicago ^t the times given here: 



DAY 


Date 


Sun Rosb 


Sun Set 


Saturday 


April 1 


5:46 a. m. 


6:23 p. m. 


Saturday 


April 8 


5:34 a. m. 


6:30 p. m. 


Saturday 


April 15 


5:24 a. m. 


6:37 p. m. 


Saturday 


April 22 


5:14 a. m. 


6:43 p. m. 


Saturday 


April 29 


5:05 a. m. 


6:50 p. m. 



Give the length in hours and minutes of each of these 5 days 
"from sun to sun," and also give in minutes the change per week 
of the days' length during April, 1905. 



224 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



10. It is noon at 12 :00. How long in hours and minutes was 
the forenoon of April 1, 1905? the afternoon? How many min- 
utes longer was the afternoon than the forenoon? 

11. Answer similar questions for Apr. 8th; 15th; 22d; 
29th. 

12. From the numbers of a common almanac compare the days' 
length of April with the same dates for October. 



Date 


Morning Twilight 


Evening Twilight 


1905 


Begins 


Ends 


Begins 


Ends 


Jan. 1st 

April 1st 

July 1st 

Oct. 1st 


5:46 a.m. 
4:10 a.m. 
2:27 a.m. 
4:26 a.ra. 


7:25 a.m. 
5:45 a.m. 
4:32 a.m. 
5:56 a.m. 


4:42 p.m. 
6:24 p.m. 
7:35 p.m. 
5:44 p.m. 


6:21 p.m. 
7:58 p.m. 
9:40 p.m. 
7:14 p.m. 



13. From the table find how long morning twilight lasted on 
Jan. 1, 1905; on Apr. 1; July 1; Oct. 1. 
» 14. Answer similar questions for evening twilight. 

COUNTING THINGS 

1. A stationer pays 25^ per dozen for lead pencils and sells 
them at 5^ apiece. What is his profit on a gross of pencils? 

2. How many are "three score and ten"? How many years are 
"a score of centuries"? How many months? 

3. Lincoln's Gettysburg address was spoken in November, 
1863. The address begins "Four score and seven years ago," etc. 
To what date did Lincoln refer? 

4. What is the value of 1 bale of paper at 12^ per quire? 

5. What is the value of 3 great gross of steel pens at the rate 
of 2 pens for 5^? 

6. What is the value of 10 great gross boxes matches at the 
rate of 2 boxes for 5^? 

§148. General Exercises. 

1. Reduce 9 great gross to units. 

2. How many great gross are there in 79,636 unitig? 



COMPOUND DEIJOAIINATE NUMBEUS 225 

3. A dealer bought 3600 lead pencils at S2.00 per gross. He 
sold them at 25 cents per dozen. What was his profit? 

4. Bought 2 gr. foot-rules at $.12 a doz. ; 7 gr. Spencerian 
pens No. 1 at $.07 a doz.; 8 gr. Eagle pencils No. 3 at $.30 
a dozen. Find the amount of my bill. 

5. A stationer bought 3 gr. boxes of paper, each box con- 
taining 6 reams. How many sheets did he buy? 

6. A stationer sold 3 quires, 20 sheets of paper to one man, 
18 quires to another, 4 reams, 15 quires to another. How many 
sheets did he sell in all? 

7. There are 2 reams, 9 quires of paper in one package, 
3 times as much in a second package, and 7 times as much in 
a third package as in the second package. How much paper 
is there in the second and third packages each? 

8. In £28 there are how many shillings? how many pence? 
how many farthings? 

9. In 18s. there are how many pence? how many farthings? 

10. In £28 18s. 9d. 3 far., there are how many farthings? 

11. In £342 15s. 6d. there are how many pence? 

12. How many pounds, shillings, and pence are there in 28,643 
pence? 

13. How many feet are there in 78 yd.? how many inches? 

14. How many inches are there in 78 yd. 2 ft. 6 inches? 

15. How many yards, feet, and inches are there in 2838 inches? 

16. How many ounces are there in 12 cwt.? in 10 pounds? 

17. How many ounces are there in 12 cwt. 10 lb. 11 ounces? 

18. How many hundredweight, pounds, and ounces in 19,360 
ounces? 

There are two classes of problems to be solved in reducing 
denominate numbers to their equivalents in different units. 

In one class numbers expressed in larger units are to be 
expressed in smaller units. This is called reduction descending^ 
or reduction from Mgher to lotoer denominations. 

In the other class numbers expressed in smaller units are to 
be expressed in larger units. This is reduction ascending^ or 
reduction from lower to higher denominations. 

Problems 15 and 18 are examples of the second class, and prob- 
lems 13, 16, and 17 are examples of the first class. 



226 BATIONAL GRAMMAR SCHOOL ARITHMETIC 

19. Reduce 160 yd. 1 ft. 9 in. to inches. 

20. Reduce 5781 in. to yards, feet, and inches. 

21. Reduce 8 bu. 2 pk. 5 qt. to quarts. 

22. Reduce 27 T. 18 cwt. 15 lb. to pounds. 

23. Reduce 23 cu. yd. 14 cu. ft. to cubic feet. 

24. Change £395 15s. sterling to dollars and cents. 

United States gold arid silver coins are .9 pure gold or silver 
and .1 copper. 

25. What weight of pure silver is there in a silver dollar 
weighing 412^ grains? 

26. The U. S. standard 5 dollar gold piece weighs 129 gr. 
What is the number of grains of pure gold in the standard five 
dollar gold piece? 

27. The 5-cent piece weighs 73.16 gr. .75 of the weight of 
the coin is copper and .25 is nickel. What is the weight of 
copper in the 5-cent piece? of the nickel in the 5-cent piece? 

28. The eagle weighs 258 gr. What is the weight of pure 
gold in the eagle? 

29. How many standard gold dollars can be coined from 1 oz. 
of pure gold? 

30. Express the following ratios : 

1 franc : $.25; 1.25 : 1 franc; 

1 mark : $.25; $.25 : 1 mark. 

31. If .52 oz. of gold is worth 43s. 4d., how many ounces can 
be bought for £35 18s.? 

32. A boy laid by a certain sum of money each week. At the 
end of 1 yr. 3 mo. 2 wk. he had saved 488.50. How much did 
he save each week? (Take 1 mo. = 4 wk. 2 days.) 

33. A man changed $350, half into English and half into 
German money. How much of each kind of money did he have? 

34. How many feet are 4 ch. 30 links? 

35. What is the ratio of 1760 yd. to 1 mi. 32 rods? 

36. The mast of a ship was 78 ft. 4 in. high. During a storm 
.3 of it was broken off. How high was the remaining piece? 

37. A four-sided field had sides of the following lengths: 
63 ch. 2 rd. ; 49 ch. 14 li. ; 53 ch. 1 rd. 16 li. and 38 ch. 24 li. 
How far is it around the field? 



COMPOUND DENOMINATE NUMBERS 227 

38. A man walked | of the length of a breakwater, which was 
1 mi. 243 rd. 5 yd. long. How far did he walk? 

39. A knot, or geographic mile, equals 6086 ft. What is 
the speed in common or statute miles per hour of a vessel that 
runs 21 knots per hour? 

40. A wheel, 12 ft. in circumference, makes how many revolu- 
tions in 1^ miles? 

41. A telegraph wire is 14 mi. 140 rd. long, and is supported 
by 386 poles, which are placed at equal distances apart, n. 
pole being at each end of the wire. How many feet apart are the 
poles? 

42. What is the cost of 18 A. 120 sq. rd. of land at $52 per acre? 

43. A man owned 14 A. of land, and sold 1428 sq. rd. How 
many acres did he have left? 

44. If 1000 shingles are needed to cover 100 sq. ft. of roof, 
how many shingles are required to cover a roof 40 ft. long and 
25 ft. wide, at the same rate? 

45. A lawn tennis court 120 ft. long and 85 ft. wide is to be 
surrounded by a strip of sod 15 ft. wide at each end and 8 ft. 
wide at each side. What will the sodding cost at $.35 per square 
yard? 

46. Find the cost of a half section of land at $45 per acre. 

47. How many square rods are there in a rectangular field 
24 ch. 45 li. long by 16 ch. 34 li. wide? 

48. How many cubic inches are there in a tank containing 
120 gal. ? how many cubic feet? 

49. How many cubic feet are there in a block of stone 5 ft. x 
4 ft. X 6 inches? 

50. How much must I pay for a board 16 ft. long and 8 in. 
wide, the board being 1 in. thick and lumber costing $35 per 
thousand? (A board foot means 144 sq. in. of surface, not over 
1 in. thick.) 

51. I bought 3 boards 12' long, 6" wide, 8 boards 16'long,and 
9" wide, and 2 boards 10' long, 12" wide and all were 2" thick; 
what did the whole cost at $30 per thousand? 

52. What is the value of a straight pile of wood 16 ft. long, 
8 ft. wide and 6 ft. high, at $7.50 per cord? 



228 RATlOKAL GRAMliAE SCHOOL AHltHMETIC 

63. What is the weight of a pile of oak boards 14 ft. long 
8 ft. 4 in. wide and 5 ft. high, at an average weight of 54 lb. per 
cubic foot? 

54. At $.75 a load of 1 cu. yd. what will be the cost of remov- 
ing a pile of earth 60 ft. long 24 li. wide and 1 yd. high? 

55. In one year the quarries of Minnesota yielded 4,000,000 
cu. ft. of sandstone. What was this worth, at $.76. a perch? 

56. Find the cost of building a stone wall 150' x 10' x 2^, at 
$3.50 a perch? 

57. A bin 12' x 8'6" x 5' is filled with wheat. What is the 
weight of the wheat at 60 lb. per bushel? 

58. What is the value of a straight pile of pine slabs 32'x 7'x 4', 
at $3.25 per cord? 

59. A grocer paid $4.50 for a barrel of vinegar (31| gal.), and 
sold it at 5^ per quart. What was his profit? 

60. A jug contained 214 cu. in. of molasses. How much did 
it lack of containing 1^ gallons? 

61. What will 1 mi. of right of way for a railroad cost at $00 
per acre, the width of the right of way being 100 feet? 

62. What will be the cost per mile for railroad ties at 1.45 
apiece, the ties being laid one every 2 ft. (Omit the odd tie.) 

63. Find the cost of the rails for 1 mi. of the road, the rails 
weighing 77 lb. per linear yard and costing $35 per long ton. 

64. Find the cost of fencing 1 mi. on both sides of the track, 
placing posts costing $.25 apiece 16 ft. from center to center, and 
using wire weighing 2160 lb. per mile, and 30 lb. of staples @ 4f 
The fence is to be 4 wires high, and labor costs 21^ per rod of 
fence. (Here count the odd post.) 

65. Reduce .865 gal. to smaller units. 

Solution.— .865 x 4 qt. = 3.46 qt. ; .46 X 2 pt. = .92 pt. ; .92 X 4 gi. 
= 3.68 gills. Am. 3 qt. 3.68 gills. . 

66. Reduce .168 gal. to smaller units. 

67. If oil is $.11 per gallon, what will be the cost of tnree 42- 
gallon barrels of kerosene? 

68. If a certain spring regularly yields 25 gal. daily, how 
many barrels of the capacity of 31^ gal. would it fill in 20 days? 



COMPOUND DENOMINATE NUMBERS 229 

69. A merchant bought 16 gal. 3 qt. of syrup at 38^ a qt., and 
sold 27 qt. for $12, 18 qt. for $9, and the remainder at 36^ per 
quart. Did he gain or lose, and how much? 

70. Beduce 2 pk. 7 qt. 1 pt. to the decimal of a bushel. 

Solution.— 1 pt.= .5 qt. 7.6 qt.= ^ pk. = .9375 pk. 2.9375 pk = 

o 

2.9375 . noAory^ u u 1 

— 3 — bu. = .734375 bushels. 
4 

71. Reduce 3 pk. 5 qt. 1 pt. to bushels. 

72. How many bushels are there in a bin 14 ft. long 7 ft. 6 in. 
wide and 5 ft. 8 in. high? 

73. A farmer sold 6 loads of corn, each load averaging 36 bu. 
2 pk. at 35 cents per bushel. What did he receive for the whole? 

74. A boy had a bushel of hickory nuts, and sold 3 pk. 7 qt. 
1 pt. What fraction of a bushel had he left? 

75. If 9 bu. of potatoes cost $4.80, what is the average cost 
per peck? 

76. A bin contained 5376.25 cu. in. of rye. How much did it 
lack of containing 3^ bushels? 

77. How many oz. of quinine will be required to prepare 12 
gross of 3-grain capsules? 

78. A coal dealer buys two car-loads of coal each weighing 
67,200 lb., at $4.50 a long ton, and sells it at $5.75 a short ton. 
What is his gain? 

THE METRIC SYSTEM 
§149. Historical. 

The metric system is a decimal system of weights and meas- 
ures adopted by the French Government soon after the French 
Revolution of 1789. The aim of the system is to base all measures 
upon an invariable standard, and to secure the simplest possible 
relations between the different units of the system. 

The unit of length, which is fundamental to the whole system, 
is called the 7neter. It was attempted to make the meter 1 ten- 
millionth of the length of the part of a meridian of the earth, 
which reaches from the equator to the pole, called a quadrant of 
the earth's meridian. The meridian of the earth was measured, 
and TTnTTArTTnr of the quadrant was obtained. A platinum bar 
equal to this length was very accurately cut and stored in the 
Government archives as the official standard of reference. 



230 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

Later measurements of the earth's meridian showed the former 
length of the meridian to be incorrect. The length of the meter 
as obtained from the erroneous measures was, however, retained, 
and the length of this bar is the standard meter. From it all the 
other units of the system are derived. 

The metric system is used for all purposes in France, and for 
nearly all scientific purposes in Germany, England, and the 
United States. 

The length of the meter is 39.37079 in., or about 1.1 yards. 

§150. Tables of Metric Measures. 

Fig. 136 shows a scale graduated along the upper edge to 16th8 
of an inch, and along the lower edge to lOOOths of a meter, called 
millimeters. 



' i 'l ' l ' I' I'I ' I'I'I ' I ' l 'I'I 'I 'I' I ' M I ' l '| 'I'| M '|' M | ' I '|M'| ' ''[ 
V a' 3' 4' 

,i.ij.iii.i.i.i,iii.u,i.i?i,iii.i.i:i.i,i,i.i!i,i,i.i.i?i.i,i,i.iii.i.i,i?i,i.i.i.iii.i.i,fl 



Figure 136 

Decimal parts of the standard units are denoted by the Latin 
prefixes, abbreviated in each case to a single, small letter: 

milli, meaning 1000th written m. 
centi, meaning 100th written c. 
deci, meaning 10th written d. 

Multiples of the standard units are denoted by the Greek 
prefixes, abbreviated in each case to a single, capital letter: 

deka, meaning 10 times written D 

hekto, meaning 100 times written H. 

kilo, meaning 1000 times written K. 

myria, meaning 10000 times written M. 

Table of Measures of Length 

10 millimeters (mm.) = 1 centimeter (cm.) = about .4 in. 

10 centimeters = 1 decimeter (dm.) = about 4.0 in. 

10 decimeters = 1 meter (m.) = about 1.1 yd. 

10 meters = 1 dekameter (Dm.) = about 32.8 ft. 

10 dekameters = 1 hektometer (Hm.) = about 328 ft. 

10 hektometers = 1 kilometer (Km.) = about .62 mi 

10 kilometers = l myriameter (Mm.) =s f^l^ut 6,21 mi 



COMPOUND DENOMINATE NUMBERS 

Table of Surface Measure 



231 



100 sq. millimeters (mm*.) 
100 aq. centimeters 
100 sq. decimeters 
100 sq. meters 
100 sq. dekameters 
100 sq. hektometers 



r= 1 sq. centimeter (cm*.) 
= 1 sq. decimeter (dm^. ) 
= 1 SQ. METER (m^.) 
= 1 sq. dekaraeter (Dm^.) 
^ 1 sq. hektometer (Hm*.) 
= 1 sq. kilometer (Km^. ) 



SQ.INCH 


SQ. 
CENTI- 
METER 





FIGUKE 137 



The cm*. = .155 sq. in. 
The m«. = 10.764 sq. ft. 
The Km*. = 247.114 A. 
1 m* =1 centare (ca.) 

Table of Land Measure 

100 centares = 1 Are (pronounded air) (a.) 
100 ares = 1 hektare (Ha.) 



Table of Measures of Volume 

The standard unit of volume is the cubic meter = 35.314 cu. 
ft. = about 1.2 cubic yards. 

1000 cubic millimeters (mm'.) = 1 cu. centimeter (cm'.) 
1000 cubic centimeters = 1 cu. decimeter (dm'.) 

1000 cubic decimeters = 1 cu. meter (m'. ) 

etc., etc. 

Table of Measures of Capacity 

The standard unit of capacity is the liter (leeter). It is equal 
to 1 cu. decimeter, and equivalent to .908 dry quarts. 

10 milliliters (ml.) := 1 centiliter (cl.) 



10 centiliters 
10 deciliters 
10 liters 
10 dekaliters 
10 hektoliters 



= 1 deciliter (dl. ) 
= 1 liter (1.) 
= 1 dekaliter (Dl.) 
= 1 hektoliter (HI.) 
= 1 kiloliter (Kl.) 



Table of Measures of Weight 

The standard unit of weight is the gram, which is the weight 
of 1 cu. centimeter of distilled water at its temperature of greatest 
density (39.1° F.). 



232 



RATIOJ^AL GRAMMAR SCHOOL ARITHMETIC 



10 milligrams (mg. ) : 

10 oeotigrams 

10 decigrams 

10 grams 

10 dekagrams 

10 hektograms 

10 kilograms 

10 myriagrams 

10 quintals 



: 1 centigram (eg.) 
: 1 decigram (dg.) 
: 1 gram (g.) 
: 1 dekagram (Dg.) 
: 1 hektogram (Hg.) 
: 1 kilogram (Kg.) 
: 1 myriagram (Mg). 
: 1 quintal (Q.) 
: 1 metric ton (T.) 



1 centigram = .15432 grain 
1 gram = 15.432 grains 

1 kilogram = 2.20462 lb. 
1 metric ton = 2204.621 lb. 



§151. Metric and TJ. S. Equivalents. 

The equivalents will be of assistance in changing the metric 
to the common system of measures. 



1 m. = 39.37 in. 
lKm.= .6214 mi. 



1 mile = 1.6093 Km. 
1 yard = .9144 m. 



1 m2. = 1.196 sq. yd. 
IKm^. = .3861 sq. mi. 
lare = .0247 A. 



1 square yard = .8361 m^. 
1 square mile = 2.59 Km^. 
1 acre = 40.47 a. 



1 m«. = 1.308 cu. yd. 
1 stere = .2759 cord 



1 liter 
IHl. 



-i 



1.0567 liquid qt 
.9081 dry qt. 
^ j 2.8376 bu. 
i 26.417 liq. gal. 



1 g. = 15.432 grains 

1 Kg. = 2.2046 lb. avoir. 

1 metric ton = 1.1023 T. 



1 cubic yard 
1 cord 



= .7645 m8. 
= 3.624 St. 



61022CU. in. = 1 1. 

1 liquid qt. = .9436 1. 

Idryqt. =1.1011. 

1 bushel = .3524 HL 

1 grain = .0648 g. 

1 lb. avoir. = .4536 Kg. 

1 T. = .9072 T. 



PROBLEMS WITH THE METRIC USTITS 

1. Take a smooth lath, or plane one, lay a straight strip of 
paper beside the lower edge of the metric rule, Fig. 13G, page 230, 
and with a sharp pencil transfer the graduations from the rule to 
the strip of paper, and then from the strip of paper to the lath. 



COMPOUND DENOMINATE NUMBERS 233 

Continue the centimeter graduation marks along your lath until 
you get a meter stick (= 100 centimeters). How does this meter 
stick compare in length with a yard stick? 

2. Explain which digit in 6.8752 m. stands for m. ; which for 
dm. ; which for cm., and which for millimeters. 

3. Find by measurement the length and the width of your 
schoolroom in meters, and write the result of your measurements 
in meters and decimals of a meter. 

4. How many centimeters long, wide, and thick is your arith- 
metic? 

5. Express the length of your pencil in centimeters. 

6. A sidewalk is 2112 m. long, how many kilometers long is it? 

7. What is the cost of 158 cm. of ribbon at $.45 a meter? 

8. How many steel rods, each 16 cm. long, can be cut from 
a rod 7.68 m. long? 

9. From a piece of wire, 98.52 m. long, a piece .047 Km. was 
cut. How long was the remainder? 

10. A bicycle track is .807 Km. long. How many meters 
would one go in riding around the track 5 times? 

11. Express 1 Km*, in square dekameters; in square meters. 

12. Find the number of square meters in the surface of your 
schoolroom. Express the same in square dekameters. 

13. Express the area of your desk in square decimeters; in 
square inches. How do the two compare? 

14. Using the decimeter as a measure, find the number of 
square centimeters in the surface of your geography. 

15. Express the same in square decimeters; in square inches. 

16. Find the area of the door in square meters. What part of 
an are is this area? 

17. A certain plot of ground contains 20 m*. How many square 
feet in the plot? 

18. With the aid of your meter stick, lay off a square meter 
upon the floor. Within this area, lay off a square yard. Note 
results. 

19. With the aid of your decimeter measure, dra\«^ a square 
decimeter. In the corner, draw a square centimeter. How many 
square centimeters could be drawn within the square decimeter? 



234 RATIOKAL GRAMMAR SCHOOL ARITHMETIC 

20. What is the area of a floor 8.4 m. long and 5 m. wide? 

21. A lot is 7.64 m. wide and .033 Km. deep. What is the 
area? 

22. What is the value of a piece of land 8 Dm. long and 6. 5 Dm. 
wide, at $32 an «re? 

23. How many ares in a street 1.5 Dm. wide and 2.48 Km. 
long? 

24. How many square meters in a floor 700 cm. long and 
500 cm. wide? 

25. A man had 3 pieces of land as follows: 8.4 Ha.; 3846 
m'., and 2.5 Km^. How many hektares of land are there 
in all? 

26. 20 liters equal how many centiliters? what part of a Dl.? 
of a Kl.? Find the difference between 12 1. and 12 quarts. 

27. Using your decimeter measure as a basis, model a cubic 
decimeter or liter. 

28. Using your square centimeter as a basis, model a cubic 
centimeter. What is the relation of a cubic centimeter to a 
gram? How many cubic centimeters in a liter? 

29. How many cubic meters are there in the volume of your 
schoolroom? 

30. What is the difference between an ordinary ton and a 
metric ton (tonneau)? 

31. A gram is the weight of 1 cm^. of distilled water. What 
is the weight in grams of 1 1. of distilled water? 

32. A liter of water weighs nearly 2^ lb. What is the weight 
of 6 1. of water in pounds? in grams? 

33. A package of silver weighs 2.58 grams. What is its 
weight in grains? 

34. How many pounds in 1 Q.? in 6 quintals? 

35. How many 2 gr. capsules will 5 g. of quinine fill? 

36. What would be the cost of the quinine at 10^ per dozen 
capsules? 

37. What is the cost of 5500 Kg. of coal at $6.50 per ton? 

38. What is the cost of 2 Q. of sugar at $.08^ a kilogram? 

39. If a book weighs 3.2 Dg., how many such books will weigh 
1.792 kilograms? 



PERCENTAGE AND INTEREST 235 

40. A box contains 2500 packages of quinine, each package 
weighing 1.25 g. How many hektograms does the contents of 
the box weigh? 

41. A vesse}, 18 cm. long, 14 cm. wide, and 22 cm. high, is 
filled with lead, which weighs 11.35 times as much as water. 
What is the weight of the lead? (1 cu. ft. of water weighs 62.5 lb.) 



PEBCEVTAGE AND IVTEBEST 
§152. Percentage. 

ORAL WORK 

In a city block there are 100 houses. 50 of them are stores; 
40 are dwelling houses; 5 are warehouses; 3 are hotels, and 2 are 
banks. 10 of them are built of brick; 30 are stone, and 60 are 
frame houses; 35 of them are 3 stories high, 25 are two stories, 
and the rest are one story high. 

1. What part of the houses are stores? How many hundredths 
of the houses are stores? 

Note. — "How many per cent?" "What per cent?" means "How 
many hundredths?" 

2. How many per cent of the houses in the block are stores? 

3. i equals how many per cent? 

4. What part of the houses are 2 stories high? What per cent 
of them are 2 stories high? 

5. What per cent are warehouses? hotels? banks? dwellings? 

6. What per cent are brick? stone? frame? 

7. What per cent are 3-story buildings? 1-story buildings? 

8. What part of the houses are warehouses? ^V equals how 
many per cent? 

9. What part of the houses are dwellings? f equals how many 
per cent? 

10. What part of the houses are frame houses? | equals how 
many per cent? 

A sidewalk tOO rd. long is made up of 10 rd. of stone walk, 25 
rd. of cement walk, 40 rd. gravel walk, 50 rd. board walk, and 75 
rd. cinder walk. 

11. What per cent of the entire walk is stone? What per cent 
cement? gravel? board? cinder? 



236 RATIONAL GRAMMAR SCHOOL .ARITHMETIC 

12. What per cent of the whole was the stone and the cement? 
the cement and the cinder? the gravel and the board? the stone 
and the board? 

In a certain nursery there are 100 oak, 75 ash, 50 maple, 30 
elm, 25 hickory, 15 poplar, and 5 walnut trees. 

13. What per cent of the trees were oak? What part were oak? 

14. What per cent of the trees were ash? What part were ash? 

15. What per cent of the trees were maple? elm? hickory? 
poplar? walnut? 

16. $2 equals what part of $8? 9 bu. equals what part of 
12 bu.? 8 hr. equals what part of 24 hr. $6 of $9? 

17. To how many hundredths of a number are the following 
fractional parts of that number equal: 

i? i? I? I? V f? F F I? t\^ tV tV irV? A? 2V 

18. Express the following as hundredths : 

3. 9. T.fl.1.6. 1.5.1.3.6. 3 
T(r> ¥7> lfJ59 tlTj €i 6> \J9 Tf> t» «> 8> T15- 

Definitions. — The words per cent mean hundredth, or hundredths. 

The sign, " %," is a short way of writing percent, or hundredths; 
thus, 2%, 8%, 33J%, mean xJo. 185* fii. and are read, "2 per cent," 
"8 per cent," "33} per cent" 

The number (as 2, 8, 33} ), written before the sign, " %," is called the 
rate per cent. 

It is well to recall that we have the foUoi^fing ways of writing such 
numbers as 6 per cent: (1) 6 per cent; (3) 6% ; (3) 6 hundredths; (4) jSq ; 
and (5) .06. The sign, " %," is merely an abbreviation for ^J^, or .01. 

19. Of 100 kinds of animals of a neighborhood, 27 die, 18 
migrate and 10 hibernate as winter comes on. What per cent die? 
migrate? hibernate? 

20. 10 kinds store food, 20 remain and feed abroad and 10 
kinds appear only in winter. What per cent store food? feed 
abroad? appear only in winter. 

WRITTEN WORK 

Review §80, pp. 123-125, as oral work. 

Note. — Whenever you can answer a problem orally, do so. Form the 
habit of using your pencil only when it is necessary. 

1. In a schoolroom containing 40 pupils, | of the pupils were 
girls. How many girls were there? 



PERCENTAGE AND INTEREST 237 

2. In the last problem, what per cent of the pupils were 
girls? 

3. In the month of September, 12 da. were cloudy. What per 
cent of the days were cloudy? 

4. Express the equivalents of the following fractions as per 
cents, or hundredths : 

i; I; 14;*; I;*; iV; A; A; if- 

5. In a sample of soil weighing 37 oz., 18 oz. were sand. What 
per cent of the soil was sand? 

6. A sample stick of timber, weighing 19 lb., contained 2 lb. 
of water. What per cent of the weight of the timber was due to 
the water it contained? 

Solution.— (1) Annex zeros after the numerator and divide thus: 
19 )2.00 But .IItV = lliV%- Why? 

7. What per cent of a yard is 1 ft.? 9 in.? 15 in.? 28 in.? 
35 in.? 1 inch? 

8. The first number of these pairs is what per cent of the 
second: 

(1) 8 of 150?' (4) ^ of 60? (7) 32.91 of 263.28? 

(2) 13 of 700? (5) 27i of 600? (8) i of i? 

(3) 7i of 12? (6) 38.45 of 769? (9) | of 12J? 

9. Change the following per cents to their fractional equiva- 
lents (fractions in their lowest terms) : 

5%; 8%; 8^%; 12^%; 12%; 16%; 16|; 22^%; 67i%; 87^%. 

10. Change to their decimal equivalents the following: 

7% ; 6i% ; 83i% ; 87^% ; 1*% ; 2i% ; 3 % ; yV of 1% ; jh of 1%. 

11. At the beginning of a school year, the lung capacity of a 
boy was 161 cu. in. By the middle of the year it was «7% larger. 
How many cubic inches had his lung capacity increased? 

12. The lung capacity of a boy at the beginning of the year 
was 160 cu. in., and at the middle of the year it was 166.4 cu. in. 
What was the per cent of increase? 



238 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

13. The standing of the several clubs in the National League 
during one season was determined from this table : 

Club Games Won Games Lost Per Cent Woh 

Pittsburg 90 49 64.7% 

Philadelphia.... 83 57 

Brooklyn 79 57 

St. Louis 76 64 

Boston 69 69 

Chicago 53 86 

New York 52 85 

Cincinnati 52 87 

Find for each club what per cent of the total number of games 
played were won. Garry results to the first decimal place, as 
shown for Kttsburg. 

14. What per cent of the farm, Fig. 4, p. 7, is the cornfield? 
the wheatfleld? the meadow? the south oat field? the pasture? 
the lot occupied by the house and grounds? 

15. 60% of the value of a mill is $5400; what is the value of 
the mill? 

16. 3% of a school of 1200 children were absent; how many 
children were absent? 

17. I pay 12% of the value of the property I occupy as rent 
every year. My rent is $240 a year; what is the value of the 
house? 

18. A house was damaged by fire, and an insurance company 
paid the owner $840 damages, which was 40% of the original cost 
of the house ; what was the original cost? 

§153. Algebra. 

ORAL WORK 

1. 50 equals what per cent of 100? of 200? of 150? of 500? 

2. 2 equals what per cent of 4? of 8? of 6? of 20? 

3. Any number equals what per cent of a number twice as 
large? 4 times as large? 3 times as large? 10 times as large? 

4. X equals what per cent of 2.r? of 42;? of 3a;? of 10a;? 

5. 7a equals what per cent of 14a? of 28a? of 21a? of 70fl? 

6. 13 equals what per cent of 17? of 28? of 45? of 55? 

13 13 X 100 A^?A 76^8 
SOLUTION.- -j^ = ^^^^^ = ^ = _X^ = 76ft %. 



PERCENTAGE AND INTEREST 239 

7. 13a; equals what per cent of 17a;? of 28a;? of 4:5a;? of 55a;? 

8. X equals what per cent of y? of z? of m? of ^? 

Note— — =-^ = 100—%. 
1^ 100 1^ ^ 

WRITTEN WORK 

1. What is 2% of $200? of $375? How is 2% of any number 
of dollars found? 

2. How is 2% of $a found? What is 2% of la? of $J? of $x? 
of $6x? 

3. How is 6% of any number found? What is 5% of the 
number a? of b? of x? of z? of 82;? 

4. How is 12J% of any number found? What is 12^% of 
a? of X? of 10a;? 

5. IIow is any per cent of a number found? 

G. How can you express a as hundredths? x, as hundredths? 
9a;? 12y? 4:5z? 

7. Express these numbers as hundredths: 16; 29; a; x; m; 
16a;. 

8. What is r% of 160? of 350? of a? of ^? of C? of 27W? 

Definitions. — The result of finding a given per cent of any amount, 
or number, is called the percentage. The amount, or number, on which 
the percentage is computed is often called the base. 

9. Calling the percentage, jh the rate per cent, r, and the 
base, 5, show by an equation how to find jy, from b and r. 

10. Replacing the symbols in your equation by the words for 
which they stand, translate the equation into the common lan- 
guage of percentage. This translation, properly made, is the 
fundamental principle of percentage. 

Principle. — The percentage equals the product of the base and 
rate divided by 100, or more briefly, 

(I) i'=r^,- 

All of percentage is contained in this equation, called a 
formula, because it formulates {expresses in brief form) a law. 

Multiplying both sides of the formula by 100, we have 

{A) 100i?-ir. 



240 RATIONAL GRAMMAR SCHOOL ARITHMETtC 

Now divide both sides of this equation by r, and write the 
second number (see §69, p. 97) first, and obtain (note that 

(II) b = lME. 

11. Translate (II) into words. How would you find the base 
(b) if the percentage (p) and rate (r) were given? 

12. Divide both sides of (A) by b, write the second number 
first, and make a rale for finding r when p and d are given. 

§154. Oain and Loss. 

1. A merchant paid $16 for a suit of clothes. At what price 
must he sell the suit to gain 25 per cent? 

2. A stationer pays $1.80 a dozen for blank books, and retails 
them at a profit of 66f %. At what price per book does he sell 
them? 

3. A grocer buys eggs at wholesale at 10^ a doz.., and retails 
them at a profit of 30%. Supposing there is no loss due to break- 
age or spoiling, at what price per dozen does he sell them? 

4. Allen paid $1 for a sled, and sold it at a loss of 20%. How 
much did he receive for the sled? 

5. From a box containing 200 oranges, 80% were sold in one 
day. How many oranges were sold? 

6. What is the percentage on $640 at 20%? at 28%?? at 35%? 
at 6i%? at 12i%? at 87^%? 

7. A farm, costing $4400, was sold at a gain of 18% ; for how 
much did the farm sell? 

8. A farm sold for $5192, which was 18% more than was paid 

for it. How much was paid for it? 

Solution.— The 18% here means 18% of what was paid for the farm 
(the cost price). $5192 is then equal to what per cent of the cost? The 
statement may be written thus: 

1.18 times the cost price = $5102, 

or, more briefly, thus: 1. 18a? = 85192. Hence, .r =—^^70. Divide and 

find the value of x. 

9. A center fielder threw a ball 90 ft., which was 12% farther 
than the third baseman threw it. How far did the third baseman 
throw it? 

Suggestion. --First answer the question, 12% of what? 



PERCENTAGE AND INTEREST 241 

10. A boy bought oranges at the rate of 5 for 3jJ, and sold 
them at the rate of 3 for 5^ ; what was his rate per cent of gain 
or loss? 

11. A merchant in shipping 150 crates of eggs lost 25 crates 
by freezing. What per cent did he lose? 

12. Hats, costing $2.75, sold for $3.50. What was the per 
cent of profit? 

13. An automobile, costing $750, sold for $1250. What was 
the per cent of profit? 

14. In a certain battle, 32,000 men were engaged, and 35% of 
all engaged lost their lives. How many lost their lives? 

15. A horse, costing $88, sold at a loss of 40%. For how 
much did the horse sell? (40% of what?) 

16. I paid $35 for a bicycle, and sold it the next season for 
60% less than I paid for it. What was the selling price? 

17. A cow gave 13 lb. of milk Nov. 1, and 15.6 lb. on Nov. 15. 
What was the rate % of gain in 14 days, in her daily yield of milk? 

18. Make similar problems on the table, page 16. 

19. The increase in population in one year in a certain town 
was 2000, which was 6J% of the population at the end of the 
year? What was this population? 

20. The death rate the same year was 1.8% of this popu- 
lation. How many deaths occurred? 

21. A man lost 27% of the books of his library by fire. He 
lost 540 books. How many books did his library contain before 
the fire? 

22. Anthracite coal cost $8.50 a ton in Nov., and rose 15% in 
1 mo. What was the price of anthracite coal after the rise in 
price? 

23. I sold a lot for $600, at a loss of 20%. What did the 
lot cost? 

24. Find the cost of a piano which sold for $187.50 at a loss 
of 25%. 

25. I bought a tennis outfit for $35, and sold it at a loss of 
15%. Find the selling price. 

26. Carpet, marked at $1.25, sold at a reduction (discount) 
of 6f %. Find the selling price. 



242 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

27. A man lost 20% of his money and after losing 10% of the 
remainder had $3,600 left. How much had he at first? 

28. A dealer sells a cask of 30 gal. of oil, and receives $30. 
In delivering it 4.5 gal. were spilled. What per cent was spilled, 
and how much should the dealer pay back? 

29. Goods damaged by fire were sold at a loss of 40%. The 
amount received was $555. What was the original cost? 

30. A suit of clothes was marked at $45, which was 50% moi-e 
than the cost. It sold at a reduction of 20% from the marked 
price. What was the per cent of profit on the first cost of the 
suit? 

§155. Meteorology. 

The number of clear, cloudy, partly cloudy, and rainy or snowy 
days for the first 4 mo. of 1903, at Chicago, are here tabulated : 

1903 CLEAR CLOUDY g^^J^ ^nSw^ 

Jan. 9 15 7 9 

Feb 10 11 7 9 

Mar 12 14 5 11 

Apr 10 13 7 13 

1. What per cent of the number of days of Jan. were clear? 
cloudy? partly cloudy? rainy or snowy? 

2. Answer similar questions for Feb. ; for March; for April. 

3. What per cent of the total number of clear days of 1903 to 
May 1st fell in Jan.? in Feb.? in March? in April? 

4. Answer similar questions for cloudy, partly cloudy and 
rainy, or snowy days. 

5. The highest velocities (speeds)in miles per hour of the wind in 
Chicago for each of the 12 mo. (beginning with Jan.) of a certain 
year were: 50; 48; 57; 70; 54; 45; 56; 47; 52; 58; 52; 58. 
What per cent of the highest velocity for April was the highest 
velocity for Jan.? for Feb.? for each of the remaining months? 

6. The total movement, in miles, of the wind in Chicago for 
each of the 12 mo. of a certain year was: in Jan. 12,736; 10,279; 
13,999; 12,820; 12,356; 10,900; 11,231; 9,839; 11,834; 13,148; 
13,583; 15,476. What per cent of the total wind movement in 
Dec. was the total movement for each of the other months? 



PERCENTAGE AND IKTEREST 243 

The table below gives the height in feet (elevation), above 
sea level and the total wind movement in miles for an entire year, 
of a number of important weather signal stations. Block Island 
is on the Ehode Island coast, and Mount Tamalpais'is a mountain 
station near San Francisco. Roseburg (Oregon) is the quietest 
place, as to wind movement, reported in the United States: 

Station Elevation Wind Movement 

Mount Tamalpais 2,375 163,203 

Block Island 26 152,838 

Chicago 823 145,193 

Cleveland 762 128,566 

New York , 314 127,267 

Buffalo 767 125,042 

Boston 125 98,755 

Philadelphia 117 95,319 

St. Louis 567 84,482 

New Orleans 51 74,299 

Louisville 525 70,396 

Washington 112 63,629 

Roseburg (Oregon) 518 30,471 

7. What per cent is the wind movement of Chicago of that of 
each of the other placed in the list? 

8. What per cent is the elevation of Chicago of that of each 
of the other places? 



9. 100 green oak leaves weighed 100 grams. When thor- 
oughly dried they weighed 39.5 grams. The dried leaves were 
then burned, and the ash weighed 2.4 grams. What per cent of 
the leaves was dry solid? mineral matter (ash)? 

10. 100 green elm leaves weighed 60 grams. 38^%, by weight, 
was dry solid, and 2f % was mineral matter. What was the 
weight of the dry solid? of the mineral matter? 

11. A school garden, 15' x 40', was seeded as follows: 25%, 
potatoes; 16|% of the remainder, peas; 20% of the remainder, 
beans; 25% of the remainder, lettuce; 33^% of the remainder, 
radishes; 20% of the remainder, parsley, and the rest in beets. 
Hot many square feet were seeded to beets? 



244 



BATIOKAL ORAHMAB SCHOOL ARITHMETIC 



12. The precipitation (rainfall) at Chicago, in inches, for 
Jan., Feb., Mar., and Apr. for 1903 was 1.09, 3.03, 1.67, and 
3.77, and the averages for these same months for 33 yr. are 2.05, 
2.11, 2.52, and 2.73. Find the rate per cent of excess, or defi- 
ciency, for each of the four months of 1903, on the 33-yr. average. 

13. Problems may be- made on the following table of data 
relating to Chicago weather: 



1902 


Temperature 


Precipitation 


Weather 


Monthly 
Mean 


Mean for 
31 Years 


In Inches 


for 3S Yr. 


Clear 
Days 


Pair 
Days 


Caoudy 
Days 


Jan. . . 
Feb. .. 
Mar. . . 
Apr. . . 
May .. 
June . . 
July . . 
Aug. . . 
Sept. . . 
Oct. . . . 
Nov. . . 
Dec. .. 


25.2 

20.8 
38.6 
46.4 
59.0 
64.2 
72.4 
68.4 
60.8 
55.2 
47.0 
26.5 


23.8 
25.9 
34.3 
46.4 
56.5 
66.6 
72.3 
71.1 
64.4 
53.1 
38.5 
29.0 


.66 
1.53 
416 
2.26 
5.08 
6.45 
5.78 
1.44 
4.83 
1.45 
2.03 
1.90 


2.08 1 

2.30 

2.56 

2.70 

8.59 

3.79 

3.61 

2.83 

2.91 

2.63 

2.66 1 

2.71 


10 
12 

6 
10 
10 

7 

12 
15 

9 
14 

8 

7 


14 

9 

13 

12 

19 

16 

13 

10 

8 

10 

13 

8 


7 
7 

12 
8 
2 
7 
6 
6 

13 
7 
9 

16 



§156. The Almanac. 

Note. — In this section "length of the day" means the length of day- 
light, or the time interval from sunrise to sunset. 

1. On Jan. 1, at Chicago, the sun rose at 7 hr. 29 min., and set 
at 4 hr. 38 min., and on July 1, it ;rose at 4 hr. 28 min., and set 
at 7 hr. 39 min. The length of the day on Jan. 1 equals what 
per cent of the length on July 1? 

2. On Jan. 1, the length of the day in St. Louis was 103.8%, 
and in St. Paul 96.4%, of the length of the same day in Chicago. 
Find the day's length in each of these cities. 

3. On Julyl, in the same places, the day's lengths were, respect- 
ively, 97.9% and 102.2%, of its length in Chicago. Find the day's 
length in each of these cities on this date. 

4. Make and solve similar problems for the calendar pages of 
a common almanac. 



PERCENTAGE AND INTEREST 245 

5. On the first of Feb., of March, of April, of May, of June, 
of July, of Aug., of Sept., of Oct., of Nov., and of Dec, the 
day's lengths in Chicago were 108%, 122.3%, 137.8%, 152*6%, 
163.2%, 163.5%, 157.9%, 144.6%, 129.2%, 132.5%, and 102.6% 
of the day's length on Jan. 1. Find the day's length on the 
dates named. 

6. On vertical lines, one for each'month, plot these rates to 
scale and draw through the points a smooth freehand curve. 

7. Similar problems may be obtained from the calendar pages 
of a CQmmon almanac. 

8. From the almanac find what per cent the period from 
*'moon rises" to "moon sets" on the first of some month is of the 
same period on the 15th or 2dth of the same month. 

9. Find what per cent the time from full moon to new moon 
is of the time from new moon to first quarter; from new moon to 
second quarter, or full moon. 

§157. Geography. 

Find these per cents to one place of decimals. 

1. Eefer to the table of p. 30, and find what per cent of the 
population (1900) of your state was in elementary and secondary 
(high) schools. 

2. Compare the result of problem 1 with the corresponding 
results for any other states. 

Data for the following problems will be found on pp. 30 and 39. 

3. What per cent of the area of the United States (without 
outlying territory) is the area of the North Atlantic division? of 
the Western division? 

4. What per cent of the total population of the United States 
(continental) is the population of the North Atlantic division? 
of the Western division? 

5. Find what per cent the area and population of your state is 
of the area and population of the division to which your state 
belongs. 

6. Find the per cent of increase of population of your state 
from 1890 to 1900. 



246 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

7. Find the per cent of increase of population of the (conti- 
nental) United States during the same time. 

8. Find the per cent of increase in population from 1890 to 
1900 of the division to which your state belongs. 

9. The following table shows the growth in territory of the 
United States and the dates when the additions were made. Find 
the per cent of increase of each addition on the date of acquiring 
to 1899 inclusive. 

Acquisition i Datb '^a Ml* 

Original territory 1783 827,844 

Louisiana purchase 1803 1, 182. 752 

Florida /. . . 1819 59.268 

Texas 1845 371 ,063 

Mexican purchase 1848 522,568 

Texas purchase 1850 96,707 

Oadsden purchase 1853 45,535 

Alaska 1867 590,884 

Hawaii 1898 6,449 

Porto Rico ^ 3,600 

Philippine Islands j^ 1899 114,000 

Guam J 200 

Isle of Pines 1899 882 

10. The area in millions of square miles of the surface of the 
earth is 148.18; of this surface 108.77 is water, and 39.41 land. 
Find what per cent of the surface of the earth is water; land. 
The land surface equals what per cent of the water surface? 

The following table contains the actual lengths of the coast 
lines of the continent and also the lengths that would be needed to 
enclose them if they wero solid, with smooth outlines, also the 
ratio of low land to high land : 

CONTINENT LE^THS 

North America 24,040 

South America 13,600 

Asia 30,800 

Africa 14.080 

Europe 17,200 

Australia 7,600 



Length, 


Ratio Low to 


IF Solid 


Highland 


10,380 


6i:9 


9,030 


9J:2J 


13,780 


10^:13 


11,760 


61:141 


6.630 


4|:1J 


5,860 


3}a 



pehcentage akd ikterest 



247 



11. Find what per cent of each number in column 2 equals the 
corresponding number in column 3. What does each per cent 
mean? Which continent has the longest coast to defend in com- 
parison with its area? 

12. Express the given ratios of column 4 of low land to high 
land in per cent, and tell what the per cents mean? 

The table below contains the heights, in feet, of mountains 
and peaks of the world : 



Mt. Everest 29,002 

Aconcagua 23,910 

Chimborazo (volcano) 20,500 

Kilimanjaro Mts 20,000 

Mt. Logan 19,500 

Karakoram Mts 18,500 

Orizaba (volcano) .... 18,312 

Mt. St. Ellas 18.100 



Popocatapetl(volc.). . 17,784 

Mt. Wrangell 17,500 

Mt. Blanc 15,744 

Mt. Shasta 14,350 

Longs Peak 14.271 

Pikes Peak 14,147 

Mt. Etna (volcano) .. 10,875 

Rocky Mts 10,000 



13. The height of Pikes Peak equals what per cent of that of 
Mt. Everest? of Mt. Logan? of Chimborazo? of Mt. Shasta? 

14. Answer other similar questions on the table. 

15. What per cent of North America (16,130,269 sq. mi.) is 
drained through the Mississippi basin (1,250,000 sq. mi.)? through 
the St. Lawrence basin (360,000 sq. mi.)? the Columbia basin 
(290,000 sq. mi.)? the Colorado basin (230,000 sq. mi.)? 

16. What per cent of the length of the Mississippi River 
(4,200 mi.) is the length of the St. Lawrence (2,000 mi.)? of the 
Columbia River (1,400 mi.)? of the Colorado River (2,000 mi.)f 
Yukon River (2,000 mi.)? 

The following table gives the lengths, in miles, and the areas 
of basins, in square miles, of 24 of the longest rivers of the world: 



AREA OF 

RivEB Length basin 

Mississippi 4,200 1,250,000 

Nile 3.900 1,300,000 

Amazon 8,600 2,500,000 

Yangtzekiang . . 3.300 650,000 

Obi 8,000 1,000,000 

Yenisei 3.000 1,400,000 

Congo 3,000 1,500,000 

Niger 2,900 1,000,000 

Hoangho 2,800 890,000 

Amur 2,700 780,000 

Lena 2,700 900.000 

La Plata 2,500 1,250,000 



Arba of 

Rtveb Length qasin 

Mackenzie 2,400 680,000 

Volga 2,400 590,000 

St. Lawrence . . 2,000 360,000 

Yukon 2,000 440,000 

Brahmaputra . . 2,000 426,000 

Colorado 2,000 230,000 

Indus 2,000 325,000 

Euphrates 2,000 490,000 

Danube 1,900 320,000 

Rio Grande 1,800 225,000 

Ganges 1.800 450,000 

Orinoco 1,500 400,000 



248 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

17. What per cent of the length of the Mississippi equals that 
of the Nile? What per cent of the area of the Mississippi basin 
equals the Nile basin? 

18. Solve other similar problems on the table. 

The following table contains the areas, in sq. mi., the alti- 
tudes, in ft. (above sea level), and the greatest depths, in ft., of 
the great lakes of the world : 

Lakb Area Altitudb depth 

Superior 31,200 602 1,008 

Huron 23.800 581 700 

Michigan 22,450 581 875 

Erie 9.960 573 212 

Ontario 7,242 248 738 

Victoria 22,167 4,000 620 

Winnipeg 9,400 710 72 

19. The area of Lake Michigan equals what per cent of that of 
Lake Superior? 

20. Similarly compare the altitudes and greatest depths of 
these lakes. 

21. Solve similar problems on the table. 

§158. Commission. 

Definition. — Commission is a sum of money paid by a person or firm 
called the principal, to an agent for doing business for the principal It 
is usually reckoned as some per cent of the amount of money received or 
expended for the principal. 

1. An agent sells 250 bu. @ 80^- on a 2% commission. How 
much money was due the agent? 

2. An agent bought for his firm 1000 bbl. of apples @ *1.25, 
and received a commission of 5%. How much money did he 
receive? 

3. What is the commission on 100 baskets of peaches @ 50^, 
the rate of commission being 3%? 

4. What is the commission on the following sales: 

(1) 80 A. land @ $75, rate of commission being 5%? 

(2) City property sold @ $8500, rate of commission 4^%? 

(3) 40 bu. tomatoes @ 75^, rate of commission 10%? 

(4) 1000 bu. wheat @ 95^, rate of commission 1^%? 



PERCENTAGE AND INTEREST 249 

(5) 100 doz. eggs @ 25 jJ, rate of commission 12%? 

(6) 1000 bu. green vegetables @ 90^, rate of commission 12^% ? 

(7) $500 collected debts, rate of commission 15%? 

(8) $1200 bad debts collected, rate of commission 20%? 

5. Find the commission on $1850 at 2^% ; at 3|% ;. at 12^%. 

6. A principal sent his agent $2652 to be used for the firm 
after deducting (subtracting) the agent's commission of 2%. How 
much money was used for the firm? 

7. A clothing house sent its agent $1224, which included a 
sum of money to be invested and 2% commission on this sum for 
the agent. What sum was to be invested? 

8. 5% commission on a certain ainount of money was $684.20. 
What was the amount? (Statement: .05a; = $684.20. Find a;.) 

Definition. — A shipment of goods sent to an agent to be sold is called 
a consignment. 

9. A consignment of 4560 bu. of wheat was sold by an agent 
at 78f^ per bushel. What was the agent's commission at 1| per 
cent? 

10. A commission agent sold the following consignment of 
goods: 20 doz. eggs at 14i^; 40 lb. creamery butter at 21^; 36 
lb. cheese at 13:^^; 80 lb. chickens at 12^^; 8 doz. live chickens 
at $3.75; 4 bbl. apples at $3.25; 16 boxes oranges at $2.25; 4 
boxes oranges at $2.50; 12 boxes grape fruit at $2.75; 25 bunches 
bananas at $1.25; 12 boxes lemons at $2.75; 16 bbl. potatoes at 
$4.25. Find the agent's commission at 6^ per cent. 

11. An agent's commission at 2|% on a certain collection 
amounted to $97.16. What was the amount of the collection? 

§159. Trade Discount. 

Definition. — A discount is a certain rate per cent of reduction (sum 
thrown off) from the prices at which articles are listed or quoted. The 
discount is usually allowed for cash payments or for payment within a 
specified time. 

1. A retail merchant buys silk at $1.20 a yard. He is allowed 
a discount of 10 ^fc for cash. He pays cash. How much does the 
silk cost him? 

Solution.— $1.20— 10% of $1.20 = $1.08. 



250 RATIONAL GEAMMAR SCHOOL ARITHMETIC 

2. A publisher sells 60 books at $1.50 and allows 20% dis- 
count. How much does he receive for the books? 

Merchants and manufacturers often publish expensive catalogues 
containing their price lists of articles, whose prices change rap- 
idly. When prices fall, instead of publishing new lists, they mark off 
an additional discount. For example, the price of a certain article 
may be catalogued thus: $60, discount, 20%, 10%, 5%. This 
means a 20% reduction, then a 10% reduction on the reduced 
price, and then a 5% reduction on the second reduced price. It 
, would be computed thus : 

$60 $48 $43.20 

.20 .10 .05 



$12.00 $4.80 $2.1600 

$60 -$12 = $48. $48 -$4.80 = $43.20. $43.20 -$2.16 =$41.04. 

Note.— Notice this is not the same as a discount of 20% + 1^?^ + 5^ 
(= 35% ). Wherein is it different? 

3. A New York merchant sells to a customer goods marked 
thus: Price, $3500; disc't, 10% 60 da. and 5% for cash. What 
must the customer pay to settle the bill by cash payment? 

Note. — The customer gets the benefit of both discounts. 
What would the customer have paid if he had been given a single 
discount of 15 per cent? 

4. A bill of plumber's supplies was marked thus: 

Price: $7.50, disc't. 20% and 7% and 5%. 
How much did the supplies really cost? 

5. Compute the amount of money needed to settle the follow- 
ing bills if paid in cash or within the shortest time mentioned: 

(1) $45; discount 25% 60 da. and 10% 5 days. 

(2) $180; discount 16|% 30 da., and 10% 10 da., and 5% cash. 

(3) $1800; discount 30% and 10%, and 7% cash. 

(4) $54; discount 12^%, and 8%, and 5% 30 days. 

(5) 46 T. coal at $5.75; discount 10% and 2% cash. 

(6) 50 men's suits at $18; discount 20% and 5% 10 days. 

(7) 4 gross tablets at 50^ per doz. ; discount 20% and 7% cash. 

6. To what single rate of discount is a discount of 20% and 
6% equivalent? 

Suggestion.— Take a base of $10Q. 



PERCENTAGE AND INTEREST 251 

§160. Marking Goods. 

In marking his goods a merchant nses the key word "harmo- 
nizes." Rewrites the seUinff price shove a horizontal line, and 
the cost price helow it J using the letters h-a-r-m-o-yi-i-z-e-s in order 
for the digits 1-2-3-4-5-6-7-8-9-0. For example a book sells at 

J>2.50 and costs $1.75. The mark would be t-t- . 

hio 

1. Using the key "harmonizes," interpret the following cost 
marks and find the per cent of profit for each cost mark : 

(1) ^; (2) i^, (3) ^-^, (4) Vm, (5) ^. 

^ ' ma ^ ^ ' or ^ ^ ^ hio ^ ' rss ^ ' rrn 

2. Complete these marks so that the selling price may be 40% 
greater than the cost price : 

(5)f^; (6)^^^; (7)^^. 
^ ' hao ^ ' rao ^ ' mos 

3. Articles bearing the following selling marks are marked to 
sell at a profit of 33^% ; fill in the cost mark, using the same key 
as above : 

(1) ^; m 4^; (3) ^; (4) ^; (5) '^', (6) ^^ 

4. Using the same key, mark articles costing the following 
prices to sell at a profit of 37^% : 

(1) 40^; (2) 72jf; (3) $1.68; (4) $2.88; (5) $6.40; (6) $8.24. 

5. Supply complete cost marks for articles sold at 50% profit 
the selling prices of which are as follows: 

(1) 90^; (2) $1.50; (3) $2.50; (4) $3.66; (5) $7.50; (6) $8.10o 

6. Solve problems 4 and 5, using the key *'black horse." 

7. Choose a key word and with it solve problems 4 and 5. 

8. A merchant wishes to mark his goods so that he may drop 
10% below the marked price and still make 20% of the cost price. 
Using the key "harmonizes," how must he mark articles costing 
the following prices : 

(1) 50^? (2) 60^? (3) $1.20? (4) $2.50? (5) $6.40? 



252 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

§161. Interest. oral work 

Review §81, pp. 125-26. The method of §81 is known as the 
six per cent method. 

In reckoning interest the year is regarded as containing 12 
mo. of 30 da. each. 

1. A man is charged $0 for the use of $100 for 1 yr. What 
per cent of the sum borrowed ($100) equals the sum ($6) he 
is charged for its use? 

2. A man is charged $21 for the use of $350 for 1 yr. The 
sum charged equals what per cent of the sum borrowed? 

Definitions. — Interest is money charged for the use of money. It is 
reckoned at a certain rate per cent of the sum borrowed for each year it 
is borrowed. 

When money earns 3, 6, 7, or 10 cents on the dollar annually (each 
year) the rate is said to be3%,6%,7%,orl0% per annum (by the year) 
and the rate per cent is said to be 3, 6, 7, or 10. 

3. At 6% per annum, how much interest does $360 earn in 
1 yr.? in 3 yr.? in 4^ yr.? in 2| yr.? in t years? 

4. Make a rule for computing the interest on any sum of 
money at 6% when the time is in years. 

5. At 6% per annum, how much interest does $1 earn in 
1 yr.? in 2 mo.? in 1 mo.? in G da.? m 12 da.? in 18 days? 

6. When the time is in months, how may the interest on any 
sum of money at 6% per annum be computed? 

7. How may the interest at 6% per annum on any sum of 
money be computed when the time is given in months and days? 
in years, months, and days? 

Definitions. — ^The sum of money on which the interest is computed 
is called the principal. The principal plus the interest is called the 
amount. Since the borrower must not only pay the interest on the bor- 
rowed principal, but also return the principal, the debt he must discharge 
(pay) is the amx>unt. 

8. How long will it require any principal (say $1) to amount 
to twice its value ($2) or to double itself at 6% per annum? 

9. Give the reasons for these statements : 

Any principal at 6%— (1) doubles itself in 200 months; (2) 
earns j^^ of itself in 2 mo. or 60 days. 



PERCENTAGE A^D IKTEEESt 253 

10. Let / stand for the interest on $p at r % for t yr. and let i 
denote the interest on %p at 6% for t yr. Explain the meaning 
of the equations : 

WRITTEN WORK 

To find the Interest. 

1. What is .07 of $450? What is the interest on $450 at 7% 
for 1 yr.? for 2 yr.? for 5 yr.? for 3| yr.? for 4J yr.? for t years? 

2. What is .08 of $1250? What is the interest at 8% on $1240 
for 1 yr.? for 3 yr.? for 5}J yr.? for x years? 

. 3. $640 was on interest at 5% from July 1, 189G, to July 1, 

1900. Find the interest. 

4. $1800 was on interest at 7% for 3 yr. 7 mo. 21 da. Find 

the interest. 

Convenient FoBtis 

I. Interest computed first at the given rate. 

$1800 principal 
.07 rate 

int. for 1 yr. 
whole years 
int. for 3 yr. 
int. for 6 mo. 
int. for 1 mo. 
int. for 15 da. 
int. for 6 da. 
^58.85 int. for 3 yr. 7 mo. 21 da. 

11. Interest computed first at 6%. 

.18 = int. on 11 for 3 yr. at 6% 
.035 = int. on |1 for 7 mo. at 6% 
.0035 = int. on |1 for 21 da. at 6% 

.2185 = int. on %\ for 3 yr. 7 mo. 21 da. at 6% 
1800 







$126.00 
3 






$378.00 


} of $126.00 


63.00 


J of 
}of 


63.00 


10.50 


10.50 


5.25 


J of 


10.50 


2.10 



1748000 
2185 



393.3000 = int. on $1800 for given time at 6% 
65.55 = int. on $1800 for given time at 1% 

$458.85 = int. on $1800 for given time at 1% 

SUOOESTIONS FOR II.— 

(1) If the interest on $1 for 1 yr. is 6^, what is the interest for 3 years? 

(2) If the interest on $1 for 2 mo. is 1^, what is the interest for 7 mo.? 

(3) If the interest on $1 for 6 da. is 1 mill, what is the interest for 21 da.? 



254 RATIOKAL aftAjtfMAft SCflOOL A&ltttMETlC 

5. How much interest must I pay for the use of $600 for 1 yr. 

5 mo. 24 da. at 7 per cent? 

6. Find the amount of $300 for 2 yr. 4 mo. 25 da. at 
4^ per cent. 

7. Find the interest and the amount under the following con- 
ditions : 

(1) 1700 for 1 yr. 7 mo. 15 da. at 3 per cent. 

(2) $400 for 2 yr. 9 mo. 27 da. at 3 per cent. 

(3) $210 for 2 yr. 5 mo. 28 da. at 6 per cent. 

(4) $150 for 1 yr. 11 mo. 13 da. at 7 per cent. 

(5) $280 for 1 yr. 6 mo. 19 da. at 4^ per cent. 

(6) $3G0 for 1 yr. 4 mo. 5 da. at 4 per cent. 

(7) $260 for 2 yr. 3 mo. 11 da. at 3^ per cent. 

(8) $500 for 1 yr. 11 mo. 14 da. at 7 per cent, 

(9) $300 for 2 yr. 7 mo. 12 da. at 5 per cent. 
(10) $625 for 3 yr. 9 mo. 18 da. at G per cent. 

To find the Principal, 

8. What principal at 8% will furnish $16 interest in. 2 
years? 

Suggestion.— What interest will $1 produce at 8% in 2 yr.? How 
many dollars will yield 816 at 8% in 2 years? 

9. What principal at 8% will produce $30 interest in ^ 
years? 

10. What principal at 6% will amount to $112 in 1 yr. G 
months? 

Suggestion.— -What is the amount of $1 at C% interest for 1 yr. 6 mo.? 

11. Find the principal which will yield $61.25 interest in 3 yr. 

6 mo. at 7 per cent. 

12. Find the principal which will amount to $972.40 in 3 yr. 
2 mo. 12 da. at 4^ per cent. 

13. Make a rule for finding the principal when the rate, time, 
and interest are givren. 

14. Make a rule for finding the principal when the rate, time, 
and amount are given. 



PERCENTAGE AND INTEREST 255 

15 r Supply the correct value for the letter in each of the fol- 
lowing cases : 

Rate Time Interest Principal Amount 



(1) 


6 % 


1 yr. 7 mo. 15 da. 


$19.50 


P 


A 


(2) 


8 % 


3 yr. 3 mo. 18 da. 


$169.02 


P 


A 


(3) 


8i% 


4 yr. 7 mo. 21 da. 


/ 


P 


$998.50 


(4) 


7 % 


6 yr. 8 mo. 16 da. 


/ 


P 


$198.42 


(5) 


4 % 


2 yr. 6 mo. 18 da. 


$122.50 


P 


A 


(6) 


5 % 


1 yr. 10 da. 


$176.00 


P 


A 


(7) 


4i% 


90 da. 


$32.50 


P 


A 



To find the Rate. 

16. At what rate per cent will $320 yield $34 interest in 2 yr. 
9 months? 

Suggestion.— How much interest will $320 yield in 2 yr. 9 mo. at 1%V 
At what rate per cent then will the same sum yield $34 interest in 2 yr. 
9 months? 

17. At what rate will $780 yield $486.60 in 5 yr. 8 months? 

18. A man invested $2000 for 2 yr. 7 mo. 27 da. and received 
$2638 at the end of this time. What rate per cent of interest 
did his investment earn for him? 

19. A man bought 120 A. of land at $85 and sold it 2 yr. 
8 mo. later for $100 per A., after having received $900 in rents 
from it and having twice paid taxes on it at 75 cents per acre. 
What was his annual rate per cent of profit? 

20. Make a rule for finding the rate when the principal, time, 
and interest are known. 

21. Find the correct value to the first decimal place for the 
letter in each of the following problems: 



(1) $580.00 


1 yr. 5 mo. 


$46.50 


r% 


(2) $1280.00 


3 yr. 10 mo. 


$500.00 


r% 


(3) $798.45 


2 yr. 8 mo. 15 da. 


$258.65 


r% 


(4) $3698.50 


1 yr. 5 mo. 19 da. 


$568.75 


r% 



To find the Time. 

22. How long will it take $80 to earn $14 interest at 4% 

annually? 

Suggestion. — How much interest will $80 earn in one year at 4% ? In 
how many years then will $80 earn $14 at the same rate? 



256 RATIONAL OBAMMAR SCHOOL ARITHMETIC 

23. How long will it take $125 to earn $57.50 interest at 8^ 
per annum? 

24. At 7%, how long will it take $648 to yield $69.84? 

Note.— When the time results in decimals of a year the decimal may 
be reduced to months and days by the method of problem 65, p. 228. 

25. How long will it take $750 to yield $750 interest at 8%? 

26. How long will it take $10 to double itself at 6%? at 7%? 

27. How long will it take $975 to amount to $1225 at 5%? 

Suggestion.— What is the total interest? What is the interest on 
1975 at 5^ for 1 year? 

28. Make a rule for finding the time, T^ required for a given 
principal, P, to amount to a given sum, J, at a given rate per 
cent, r? 

29. Make a rule for finding how long it will take a given 
principal to earn a given interest at a given rate per cent. 

30. Find the time, T!, in years, months and days under the 
conditions stated in each of the following problems : 

PmNciPAii Rate Interest Time 



(1) $66.00 


7 % 


$28.60 


T 


(2) $460.00 


6 % 


$31.05 


T 


(3) $750.00 


Hlo 


$147,475 


T 


(4) $1260.00 


5 % 


$213.15 


T 


(5) $2460.00 


4 % 


$321.44 


T 



31. Supply the value for which each letter stands in the proh- 
lems of the following table : 





Principal 


Rate 


Time 


Interest 


Amount 


(1) 


WOO 


6% 


3 yr. 3 mo. 8 da. 


I 


A 


(2) 


$175.00 


h% 


4 yr. 9 mo. 15 da. 


I 


A 


(3) 


, $800.00 


7% 


T 


I 


$926.00 


(4) 


$475.00 


8% 


T 


$142.50 


A 


(5) 


$1266.00 


4% 


T 


I 


$1349.85 


(6) 


P 


6% 


5 yr. 7 mo. 27 da. 


$509.25 


A 


(7) 


$1575 30 


v% 


1 yr. 4 mo. 18 da. 


I 


$1662.461 


(8) 


$728.25 


^fo 


T 


$209,736 


A 


W 


$864.75 


rfc 


2 yr. 1 mo. 15 da. 


$69.29 


A 



PEROEKTAGE AKD INTEREST 

32. Complete, to mills, the following interest table : 
Interest Table: Principal «100. 



257 



Time 


Rate 


3% 


4% 


5% 


6% 


7% 


1 da 


$.008 


$.011 


§.014 


$.017 


$.019 




2 da 














3 da 














4 da 














5 da 














6 da 














1 mo 




1 








2 mo 




1 








3 mo 














6 mo 












1 vr 








1 






I 



33. Compute by the table the interest on $758 at 7% for 
4 mo. 12 da. ; for 7 mo. 8 da. ; for 1 yr. 3 mo. 10 days. 

§162. Algebra. 

1. Compute the interest on $750 at 5% for each of the 
following times: (1) 1 yr. ; (2) 2 yr. ; (3) G^ yr. ; (4) 31 yr. ; 
(5)25|yr.; (G) a; yr. ; (7) / years. 

2. Find the interest and the amount on $1250 for each of 
the following : 



(1) 5%, 1 yr. 

(2) G%,2iyr. 

(3) 7%, 6 yr. 



(4) 6%, 3iyr. 

(5) 4%,7fyr. 
(0) 3%, f yr. 



(7) r%, ^yr. 

(8) r%,a:yr. 

(9) r%, nyr. 



3. Denote the interest on a certain principal, P, by 7, the rate 
by r, and the time (in years) by /, write an equation showing liow 
to find I from P, r, and /, and translate into words the meaning 
of the equations. 



25B RATlOKAL 6IIAMMAR SCttOOL AlllTHMEtlC 

Solution.— 



(I) 



(1) I=Px^Xt, or 
^^) ^= 100 - ^' 



Translated into words: (1) means, ^'Interest equals the product of the 
principal, the rate divided by 100, and the time (in years)." 

(2) and (8) mean, ''Interest equals the product of principal, rate, and 
time, divided by 100." 

4. Calling A the amount, / the interest, and F the principal, 
write an equation showing how to find A from / and F, 

5. Write an equation to show how to find F from /, r, and /, 

and state in words the meaning of the equation. 

Prt 
If we multiply both sides of the equation, 1 = -jrrr, by 100 we 

have the equation, 100 J = Prt. Now to show how to find P from J, r, and 
t, divide both sides by rt, and write the second member on the left. We 
then have: 

(II) P=M_^. 

G. State in words the meaning of formula (II). 

7. Show, by proper multiplications and divisions of equation 
(I) (3), that the rate r may be found from P, /, and ty by the 
equation, 

(III) r = -^^. 

8. State in words the meaning of formula (III). 

9. Show from (I) (3) that the time t may be found from /, P, 
and r by the formula, 

(IV) t^l^ 

10. State in words the meaning of formula (IV). 

11. Solve by formulas I-IV the following problems: 

(1) F = $64, r = 8, and jf = 2^; find 7. 

(2) /= $24, r = 6, and if = H; find P. 

(3) / = $75, F = $850, and ^ = 2 ; find r. 

(4) /= $120, F = $G00, and r = 10; find t. 

(5) 7= $230,131, F = $722, and t = ^; find r. 



PERCEKTAQE AND INTEREST 269 

§163. Promissory Notes. 

Definitions. — A promissory note is a written promise, made by one 
person or party, called the maker, to pay another person or party, called 
the payee, a specified sum of money at a stated time. 

The snm of money for which the note is drawn is called tJieface value, 
or the face, of the note. 

The date on which the note falls due is called the date of maturity, 
and the time to run is the time yet to elapse before the note falls due. 

/c?5^ Kankakee, III., ^ne 3, '/903, 

Jrme/u t/auajBlier date I promise to pay to the 
order of S^, ^^M. ySadef^, ^wo S^^unt/?^ •::^V^ ^^ 



00 
100 



Dollars, for value received, with interest at ofx per cent per 
annum from date. 

1. lVh« is the maker of the above note? the payee? What is 
the face of the note? the date? the rate of interest? the date of 
maturity? the time to ru^? 

2. A promissory note, unless otherwise specified in the note, 
draws interest on its face value at the rate mentioned in the note 
from the date of the note until it is paid. Compute the interest 
and the amount on the foregoing note if it was paid Sept. 1, 1903. 

3. Find the interest on the following note, paid Aug. 30,1903: 

$875. Urbana, III, May 18, 1897. 

One year after date I promise to pay to James Black, or 
order, Eight Hundred Seventy-five and ^ Dollars, at 
Busey's Bank, for value received, with interest at the rate 
of seven per cent per annum from date. 

Due May 18, 1898. HENRY OSBORN. 

Note. — To find the time for which interest is to be computed, pro- 
ceed thus: 

Convenient Form Explanation. — 
Date of payment, 1903 2 10 2 mo. 10 da. = 1 mo. 40 da. 

Date of note. 1897 5 18 1 «io. 40 da. — 18 da. = 1 mo. 22 da. 

' 1903 1 mo. = 1902 13 mo. 

Time, 5 8 22 1902 18 mo. — 5 mo. = 1902 8 mo. 
yr. mo. da. 1902 — 1897 = 6 yr. 



2G0 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

4. Find the amount of each of the following notes : 
Face Rate Date or Note Date or Payment Interest Am'nt 

(1) $250 6% Mar. 12, 1899 Jan. 1,1902 

(2) $635 7% Nov. 20, 1896 July 15, 1900 

(3) $2400 5i% Dec. 12, 1899 Jan. 8, 1903 

(4) $3865 4i% Oct. 13, 1901 Sept. 7, 1903 

(5) $3640 Si% Aug. 17,1900 Mar, 3,1903 

§164. Discounting Notes. 

Definition. — Discount is a deduction from the amount due on a note 
at the date of maturity. 

In some cases promissory notes do not draw interest. The fol- 
lowing is an example: 

John C. Cannon purchased a self-binding harvester from A. R. 
Crow for $120 and gave him^the following note in payment: 

1120. Peoria, III, June 20, 1899. 

Eighteen months after date, for value received, I promise 
to pay to A. R. Crow, or order. One Hundred Twenty and 
^ Dollars, without interest until due. 

Due Dec. 20, 1900. JOHN C. CANNON. 

1. On S0pt. 20, 1899, Crow sold the note to Adams at such a 
price that Adams received his purchase money and 8% interest on 
it until the date of maturity (Dec. 20, 1900). How much did 
Adams pay for the note? 

Suggestions.— Any principal at 8% will amount to 110% of itself in 
1 yr. 3 mo. Why? Hence, 1120 = 110% of what number? Or, better, 
1.10a? = 1120. Find the value of x. 

Definitions.— The sum of money which, at the specified rate and in 
the time the note is to run before falling due, will amount to the value of 
the note, when due, is called the present worth of the note. The difference 
between the value of the note, when due, and the present worth is called 
the true discount. 

The bank discount of a note is the interest upon the value of the note 
when due, from the date of discount until the date of maturity. 

2. If Crow had sold the above note to a banker at a discount 
of 8%, the banker would have computed the interest at 8% on 
$120 from the date of sale (Sept. 20, 1899) until the date of 
maturity. How much would he have received for the note? 
Find the difference between the true and the bank discount of 
the note. 



PERCENTAGE AND INTEREST 261 

3. C. A. Thomas bought a road wagon of J. K. Duncan, 
giving' the following note in payment: 

165. Pekin, III, Sept. 10, 1900. 

Two years after date I promise to pay J. K. Duncan, or 
order, Sixty -five Dollars, value received, with interest at 7% 
per annum. C. A. THOMAS. 

Due Sept. 10, 1902. 

On March 10, 1901, Duncan sold this note to a bank at 7^% 

discount. How much did Duncan receive for the note? 

Note. — Remember, bank discount is computed on the amount of the 
note when due, for the time to run from date of sale. 

4. Counting money worth 7%, how much did Duncan receive 
for the wagon? 

5. A man bought a horse, giving in payment his note for 
1 yr. for 185, dated Feb. 2G, 1903, and drawing interest at 7% 
from date. Two months later 'the holder of the note discounted 
it at a bank at 6%. What was the discount? What did the bank 
pay for the note? 

6. Find the bank discount and proceeds on the following 
notes : 

FACE RATE DATE OP Note M^iTi^PX ^t'^A^?' oV^ Dis^ 



Maturity Sale 



Disc, count 



(1) $60 6 % Apr. 6. 1897 Oct. 6, 1898 June 15, 1897 7 % 

(2) $275 6i % Aug. 7, 1899 Nov. 7, 1901 Mar. 13, 1900 6 % 

(3) $350 7J% Sept. 10, 190U Dec. 10. 1902 Jan. 15, 1901 6% 

(4) $700 8 % Feb. 15, 1901 Nov. 15, 1903 June 20, 1901 6 % 

(5) $858 6 % Jan. 1. 1900 July 15, 1903 May 19, 1900 7 % 

(6) $1260 5 % Feb. 28,1896 Aug. 31, 1900 Aug. 8,1896 6 % ..... 

(7) $1800 5i% June 19, 1897 Aug. 19, 1901 Dec. 29, 1898 6% 

(8) $2450 7% Nov. .18, 1899 Feb. 28, 1902 Dec. 1, 1900 ^5^% 

(9) $3865 6 % Dec. 20, 1901 Nov. 20, 1902 Feb. 12, 1902 5 % 

(10) $8600 4J% Oct. 18,1900 Jan. 18, 1904 Apr. 2,1901 i% 

§165. Partial Payments. 

Definition. — When a note or bond is paid in part the fact is 
acknowledged by the holder by his writing the date of payment, the 
sum paid, and his signature on the back of the note or bond. This is 
called an indorsement 

Partial payments are made only (1) on notes which read, **0n or 
before, etc.," (2) by private agreement between the maker and the holder 
of the note. 



262 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

For calculating the balance due on a note or bond on which 
partial payments have been made, nearly all the states have 
adopted the following rule, known as '*The United States Kule 
of Partial Payments," which has been made the legal rule by a 
decision of the Supreme Court of the United States: 

Rule. — Fmd the amount of the principal to the time when 
the payment or the sum of the payments equals or exceeds the 
interest due; subtract from this amount the payment or the sum 
of the payments. Treat the remainder as a new principal and 
proceed as before. 



ILLUSTRATIVE EXAMPLES 

1. Find the balance due on the following note at maturity: 

$1250. 'Chicago, III. May 21, 1900. 

On or before two years after date I promise to pay to the 
Order of P. A. Hopper Twelve Hundred Fifty and ^^ 
Dollars at the Corn Exchange National Bank, for value 
received, with interest at 6 per cent per annum. 

Due May 21, 1902. JOHN P. MILLER. 

The indorsements on the back of this note were as follows: 
Nov. 21, 1900 $80.00 

Feb. 21, 1901 $10.00 

May 21, 1901 $150.00 

Feb. 21, 1902 $500.00 



PERCENTAGE AKD INTEREST 263 

Solution by Rule 

Principal on May 21, 1900 11250.00 

Interest for 6 mo. on $1 .03 

Interest due Nov. 21, 1900 $ 37.50 

Amount Nov. 21, 1900 (date of first payment of $80) §1287.50 

First payment 80.00 

New principal Nov. 21, 1900 11207.50 

Interest for 3 mo. on $1 .015 

Interest to Feb. 21, 1901 (date of second payment of |10) .... $ 18.11 

Payment being less than interest no settlement is made. 

11207.50 
Interest for 6 mo. on $1 .03 

Interest to May 21, 1901 (date of third payment of $150) $ 36 23 

Amount May 21. 1901 $1243.73 

Sum of second and third payments ($10 + $150) 160.00 

New principal May 21, 1901 $1083.73 

Interest for 9 mo. on $1 .045 

Interest to Feb. 21, 1902 (date of fourth payment of $500) . . . $ 48.77 

Amount Feb. 21, 1902 $1132.50 

Fourth payment 500.00 

New principal Feb. 21, 1902 $ 632.50 

Interest for 3 mo. on $1 .015 

Interest to date of maturity, May 21, 1902 $ 9.49 

Balance due at maturity, May 21, 1902 $ 641.99 

2. The following payments were made on a $650 note, bearing 
7% interest and dated April 20, 1901: 

July 30, 1901 $75.00 

• Jan. 15, 1902 $15.00 

Aug. 12, 1902.. ..$175.00 

Jan. 1, 1903 $50.00 

Find the amount due April 20, 1903.. 

3. A note of $2800, dated Feb. 23, 1900, and bearing 7% 
interest, carried the following indorsements: 

Feb. 23, 1901 $100.00 

July 16, 1901 $50.00 • 

Jan. 1, 1902 $800.00 

July 15, 1902 $85.00 

Nov. 28. 1902 $380.00 

Find the amount due Feb. 23, 1903. 



APPLICATIONS TO TKAK8P0KTATI0N PKOBLEMS 




bo 



^ 13 ^ 



OOT*Cf 






009*09 



00T*9C 






o 

rd 
^ 



^ 



o 



S3 

bo 

c 

o 



QQ 
5=1 
<»1 



e 

^ 






o 



0) 



0) o 
^ o 

•^ tl 

rid P 

bO w 

•^ o 
^ -^ 

O 0) 

o 




264 



APPLICATIONS TO TRANSPORTATION PROBLEMS 265 

3. Give the following distances in feet : 

(1) Between the centers of the leaders; 

(2) Between the centers of the rear leader and of the front 
driver; 

(3) Between the centers of the drivers; 

(4) Between the centers of the rear driver and of the trailer; 

(5) Between the centers of the trailer and of the front wheel 
of the tender (tank) ; 

(6) Between the centers of the front two wheels of the tender; 

(7) Between the centers of the front leader and of the 
trailer; 

(8) Between the centers of the front leader and of the trailer ; 

(9) Between the centers of the front leader and of the rear 
tender truck wheel. 

4. In the drawing the distance between the centers of the 
engine truck wheels is \y\ The actual distance between the cen- 
ters of the engine truck wheels is 85". . Find the scale of the 
drawing of Fig. 138. 

5. Using the scale found in problem 4, find by measuring with 
a ruler graduated to IGths of an inch the actual distance from 
the center of the front engine truck wheel to the front tip of the 
pilot (cow-catcher). 

6. The actual distance from the center of the rear tender truck 
to the rear end of the tender is 5' 3^". What is the actual dis- 
tance from the front tip of the pilot to the rear end of the tender? 

7. In the drawing it is If" from the top surface of the rails to 
the top of the smokestack. How high is the top of the smoke- 
stack above the rails? 

8. In the drawing, the distance from the top surface of the 
rails to the top of the boiler is 1". How high is the top of the 
boiler above the top of the rails? 

9. How high is the top of the engine cab above the top of the 
rails? The top line of the tender? The top of the lantern con- 
taining the headlight? 

10. Find the size of the windows of the engine cab. 

11. Give the distance in inches (1) between the nearest points 



266 RATIONAL GBAMMAIi SCHOOL ARITHMETIC 

on the rims of the drivers ; (2) between the nearest points on the 
rims of the leaders; (3) between the nearest points on the rims of 
the front tender wheels. 

12. IIow long are the radii of the leaders shown in the cut? 
How long are the circumferences of these wheels? 

13. Compute the radii and the circumferences of the tender 
■ wheels. 

14. Find the circumference of the drivers; of the trailers. 

15. When the drivers tiirn round 240 times a minute, how fast 
does the engine go? 

16. How many times do the leaders turn while the drivers 
turn round once? 

17. The weights written beneath indicate the number of 
pounds of the weight of the engine which is borne by the different 
pairs of wheels. Find the total weight of the engine. 

18. The weight of the- empty tender is 43,000 lb. When the 
tender is loaded it carries 10 T. coal and 7000 gal. of water. 
A cubic foot of water weighs 62^ lb., and contains 7^ gal. Find 
the total weight of the loaded tender. 

19. The engine shown in the cut drew a train of 3 sleepers, 
averaging 93,000 lb. ; 5 passenger coaches, averaging 78,000 lb. ; 
an express car, weighing 34,000 lb. ; and a mail car, weighing 
75,000 lb. What was the total weight of the train, including both 
engine and tender? 

20. The force (pull) exerted by an engine to draw a 
train is different for different speeds. For a speed of 10 mi. 
an hour it has been found that on straight, level track an 
engine must exert a force of 4f lb. for each ton of weight of the 
train, including the weight of both engine and tender. Find 
the force required to draw the train of problem 19 under these 
conditions. 

21. For a speed of 15 mi. an hour a force of 5^ lb. per ton of 
train weight is needed. What force will draw the* train of prob- 
lem 19 at this speed? 

22. Following are the forces for different speeds from 20 to 75 
mi. an hour. Find the pulling (tractive) force to be exerted by 



APPLICATIONS TO TRANSPORTATION PROBLEMS 



2G7 



the engine to draw the train of problem 19 at each indicated 
speed : 



Speed 


Force in Lb. 

PERT. 


Tractive 
Force 


Speed 


Force in Lb. 

PER T. 


Tractive 
Force 


20 


6i 




50 


lU 




25 


.n 




55 


12; 




30 


8 




60 


13 




85 


B| 




65 


131 




40 


9 




70 


14i 




45 


lOi 




75 


15J 





23. Find the difference between each number and the number 
next above it in the table. What do you find? 

V 

24. In the equation F= -^ + 3, let F stand for the force in 

pounds per ton and let V stand for the speed in miles per hour. 
Let F = 20 in the equation. Find F by dividing 20 by 6 and 
adding 3 to the quotient. Compare your result with the number 
of column 2, and in line with 20. What do you find? 

25. Let V = 25. Find F and compare with the number of the 
table m line with 25. 

26. Let V equal other numbers of columns 1 or 4 and find F 
for each speed. 

Definitions. — Replacing V in this way by numbers like 20, 25, and 
so on is called substituting for V the numbers 20, 25, and so on. 

Performing the operations indicated in the equation and obtaining 
the nmnber for F is called find- 
ing the value of F. 

27. How does the equa- 
tion say that the force in lb. 
per ton {F) needed to draw 
the train can be obtained 
when the speed (F) of the 
train is known? 

28. The speeds of the 
table (problem 15) are plotted 
to scale on the horizontal line 
of Fig. 139, and the forces, in lb. per ton of load, are plotted to a 
different softle on the vertical parallels. The points X to 12 are 



8 



5^ 



4?' 



20 25 30 35 'Mo 45 50 55 eio 65 70 7S 80 85 M 

Speed (V) in mi. per hr. Scale /fl"i Smi.per hr 
Figure 139 



268 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

the upper ends of the vortical lines, whose lengths represent the 
successiye numbers of columns 2 and 5 of the table. Place the 
edge of a ruler along these points, or stretch a thread taut just 
over them. How do the points seem to lie? 

29. Notice the horizontal and the vertical scales and make a 

drawing like that of Pig. 139 to a scale 4 times as large. 

Y 

30. Assuming that the same law, /'= — + 3, relating force and 

speed, holds also for a speed of 80 mi. per hour, substitute F= 80 
in the equation and compute F, the force in pounds per tonneeded 
to draw the train 80 mi. per hour. Make a similar computation 
for F= 85 and for F=90. 

31. Plot to scale on the 80, 85, and 90 lines of your enlarged 
drawing the computed values of F^ these lines becoming 13, 14 and 
15. Be careful to get each computed, F, on the proper vertical line. 

32. Stretch a string, or place a ruler, along the points 1 to 15 
of your drawing. Do the points added from your computed 
values seem to lie on the straight line through the points 1 to 12? 
With a ruler draw a single straight line through all the points of 
your drawing. 

33. Mark a point on the horizontal line midway between the 
35 and 40 points. What speed does this point represent? Draw 
a vertical from this point np to the line through the points 1 to 
12. Mark the upper end of this vertical a. What force in 
pounds per ton does the line from a to 37^ represent? 

34. How could you find from the drawing the number of 
pounds per ton needed to draw the train 42^^ mi. per hour? 
67^ mi. per hour? 22^ mi. per hour? 15 mi. per hour? 17^ mi. 
per hour? 10 mi. per hour? 5 mi. per hour? 

35. How could you find from the equation, i^= — 4-3, the 

numbers of pounds per ton needed to draw the train at the speeds 
18, 36, 42, 57, and 69 mi. per hour? Compute these forces and 
compare them with the numbers found from measurements oi the 
drawing of Fig. 139. 

Remark.— The straight line through the points 1 to 12 is said to 
represent the equation, F=-^ -\- d. 



APPLICATIONS TO TRANSPORTATION PROBLEMS 269 

§167. Laws for the Drawing of Loads. 

1. The force (F), say 200 lb., needed to draw a load (L), say 
KOO lb., on a common road wagon over loose sand is given by the 
law F= iL, or F= .25Z. Find F for these loads. 

(1) Z = 864 1b.; (2) i = 1280 lb. ; (3) i = 2G48 1b.; (4) i = 
32G8 lb. ; (5) Z = 4893 lb. 

2. Over fresh earth, the law is F= .12^Z. What forces {F, 
in pounds) are needed to draw the following loads, in pounds, 
over fresh earth: 

(1) Z = 680? (2) i = 1624? (3) Z = 2160? (4) i = 3840? (5) Z' 
= 4580? 

3. With common road vehicles on dry level highways the law 
of pulling (tractive) force (in pounds) h F= .025 i. L denotes 
the combined weight of the wagon and load in pounds. What 
forces will be needed to draw the following loads : 

(1) 45 bu. wheat on a 1200-lb. wagon? (2) 2 T. coal on a 2200- 
Ib. wagon? (3) 1^ T. hay on a 1680-lb. wagon? (4) A traction 
engine weighing 8^ T.? (5) A thresher, weighing 4^ T. 

4. On well packed gravel roads the law is F= .052 L. To 
draw a certain load on gravel road a tractive force of 44.72 lb. was 
exerted. What was the load L? 

Solution. — Dividing both sides of the equation by .052, we have 

IT IT 

-7— = i, or, what is the same thing, L = -T^p^r. We have then, by sub- 

44 72 
stituting, L = ^^~- = 860. Ana. L = 860 lb. 

5. Under the conditions expressed in problem 4, find the loads 
the following forces will draw : 

(1) i^.= 54.08 lb.; (2) /'= 66.56 lb.; (3) 7?^= 137.8 lb.; (4) 
201.76 1b.; (5) 234 pounds. 

6. The law of tractive force, on good, straight, level railroad 
track is ^= .0035 Z; on fair track, straight and level, it is F = 
.0059 L. Weights of cars are givefl in problems 15 and 17, p. 129, 
and in problem 19, p. 2G6. Make and solve problems based on 
these facts. 



270 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



7. The tractive force (F) that can be exerted by a locomotive 
on dry track, straight and level, is about .3 of the part of the 
weight of the locomotive which rests on the drivers. The loads 
that c^n just be moved along by a locomotive are given by the 
equations of the last problem. Make and solve problems based 
on the following actual weights upon the drivers of certain loco- 
motives: 

(1) 80,890 lb. ; (2)*feo,850 lb. ; (3) 86,030 lb. ; (4) 106,875 lb. ; 
(5) 112,190 1b.; (6) 131,225 1b.; (7) 14:1,320 1b.; (8) 202,232 lb. 
(See also Fig. 139). 



8. Problems may also be made on 


the following laws for road 


wagons: 




(1) Broken stone (fair) 


. . ii^=.028 i; 


(2) Broken stone (good) . 


• jP=.015 L; 


(3) Worn macadam . . 


. . jP=.033i; 


(4) Nicholson pavement 


. . F= .01^9 L; 


(5) Asphalt pavement . 


. . F^.012 L; 


(6) Stone pavement . , 


. F=.019 L; 


(7) Granite pavement . 


. . jP=.008i; 


(8) Plank road . . . 


. . F^.OIOL. 



9, Find the value oif a; in these equations : 

(1) .35a; = 7 ; (2) .58a; = 11.6 ; (3) 1.28a; = 3.84 ; (4) .092a; = 3.68 ; 
(5) .019a; = 133; (6) 7.51a; = 302.8; (7) •175a; = 10.5; (8) 1.093a; = 

13.116. 

7 
Solution of (1): a? =-5-= 20 

.oD 

10. Writing law (1) of problem 8 in the form .028Z = F, find 
the loads (L) to 2 decimals, the following, forces (F) will draw 
under the conditions of problem 8 (1) : 



(1) 50 lb. ; 

(2) 135 lb. ; 



(3) 56.28 lb. 

(4) 165i lb. ; 



(5) 68.58 lb. ; 

(6) 110.35 lb. 



11. To find the loads (i) th&t given forces {F) will draw, the 
laws of problem 8 are more conveniently written^thus : L = ,j^^^ F= 
35.71 F. Change other laws of problem 8 to this form. 



13, 


, Fii 


idtl 


le load, 


of^: 










(1) 


i?'= 


66 1b. 




(2) 


F^ 


363 lb. 


14, 


. Fiud L in F= 




(1) 


F= 


120 lb. 




(2) 


F= 


600 lb. 



APPLICATIOIS^S TO TRANSPORTATION PIIOBLEMS 271 

12. Find the load, />, in Z = 35.71 F, for each of the following 
values of the force F: 

(1) i^=2801b.; (S) F= 78.6 1b.; (5) i^= UO lb. ; 

(2) i^=145lb.; (4) i^= 175.8.1b. ; (0) /^= 700 lb. . 

1 F* 

Ly in L = -xg^ FyOT L= a, for these values 

; (3) F^ 132 lb. ; (5) F= 26.4 lb. ; 
; (4) F= 528 lb. ; (6) F= 52.8 lb. 
= .012 L for the following values of F. 
; (3) F= 720 lb. ; (o*) F = 840 lb. ; 
; (4) i^= 480 lb. ; ((>) E = 84.48 lb. 

15. Find a; in y = .325 x for each of the following values of y: 

(1) y = 65; (4) ?/ = 29.25; (7) 7^ = 526.5; 

(2) y = 97.5; (5) y = 58.5; (8) y= 10.53; 
(3)*y = 48.65; (6) y = 17.55; (9) ?/ = 357.5. 

16. On fair track, straight and level, the force which an 

engine can exert to pull a load is about j*^ of that part of its 

weight which is carried by the drivers. Show that this may be 

written as an equation thus : 

4 

F=^ — W. 

15 

Tell what the letters stand for. 

17. By the aid of the law F-^ ^^ W, find the pulling force of 
engines having the following weights on the driving wheels: 

(1) r= 105000 lb.; (4) W= 99000 1b.; (7) W= 87300 1b.; 

(2) W^ 90000 1b.; (5) W^= 107200 lb. ; (8) }r= 171300 lb. ; 

(3) }r= 101500 lb.; (6) .PF= 121500 lb. ; (9) ir= 136350 lb. 

18. Show how to change F=^^ ir first into 15 F= 4 irand 
then into Tr= 3.75 F, The letters mean the same here as in 
prob. 12. State the law W= 3.75 2^ in toords, 

16. Find the weights on the driving wheels of engine^ that 
can exert the following pulling forces: 

(1) i^= 28000 lb. ; (3) F= 26400 lb. ; (5) 37600 lb. ; 

(2) F= 30400 lb. ; (4) F= 32640 lb ; (6) 39640 lb. 
20. Find the values y in the equations if x = 138.5 : 

(1) y = 32 3;; (3) y = .036.r; (5) z/ = .0036a;; 

(2) y = 16.8 x; (4) y = .065 x\ (6) y = .0159 x. 



372 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

CONSTRUCTIVE GEOMETEY 

§168. Problems. 

Problem I. — Draw a perpendicular to a given line from a 
point upon the given line. 

Explanation.— Let AB denote the given line, and let P be a point 
upon AB. 

We wish to construct a perpendicular to AB at P. Place the pin 
foot on P and close the compass feet until their distance apart is less 

than either PA or PB. Pi is such 
3^4 a distance. Then with Pi as 

radius draw a short arc across 
AB at both 1 and ^. 

Now spread the feet a little 
wider than Pi, place the pin foot 
on 1 and draw arc S. Arc S may 
a be drawn either above or below P. 
P 12 ^ In Fig. 140 it is above P. With- 

FiGUBB 140 ^^^ change of radius put the pin 

foot on 2 and draw arc 4 across 
arc 3. Let D be the intersection of arcs S and 4- 

With the straight edge of your ruler draw the line PD, PD is the 
desired perpendicular to AB, 



'A* 



+ 



EXERCISES 

1. Draw straight lines on your paper, mark a point upon each 
line and draw perpendiculars through the marked points until you 
understand fully how it is done. 

2. Draw the perpendicular to any line you may draw on your 
paper through a marked point on the Jine, drawing arcs 3 and -4 
below the line. 

3. Draw a straight line on the blackboard, mark a point upon 
it, and with crayon, string, and ruler draw a perpendicular to 
the linor through the marked point. 

4. Draw a straight line on the ground and along the edge 
of a board, mark a point on it, and with cord, stake, and a straight 
board, draw a perpendicular through the point. 

Problem XL — Draw a perpendicular to a given line through 
a point outside of (not on) the given line. 



COKSTRtTCTlVE (^EOMETHY 



2^3 



Explanation.— Let AB denote the given line and P the marked 
point outside of AB. We wish to draw a perpendicular to AB through P. 

Put the pin foot on P and „ 

spread the feet far enough apart 
so that the pencil foot will reach 
helow the line AB. Then draw a 
short arc at 1 and at ^. 

Put the pin foot on 1 and 
draw arc S. Then with pin foot 
on $f draw arc 4 crossing 3 at D. 

Connect P and D with ruler 
and pencil. 

PD is the desired perpendic- 
ular. 

NoTE.^The arcs S and 4 might 
be drawn with radii of any length 
greater than half the distance 
from 1 tx) S; but both S and 4 
must be drawn with the same 
radius. 



"^ 



-^^ 



;c^ 



FlGUBB 141 



EXERCISES 

1. Draw straight lines on paper or on the blackboard, mark 
points outside of them, and draw perpendiculars to the lines, until 
the way of doing it is fully understood. 

2. With cord, stake, and edge of a board draw a perpendicular 
on the grotind through a point not on a line. (Fig. 37, p. 101). 

Problem III. — To draw a square inside of a circle of 
radius |^". 

C 

Explanation.— Let 0, Fig. 142, be the 
center and let OB = }i". 

Draw a diameter AB. 

Spread the compass feet apart wider 
than {i" and putting the pin foot first on A, 
\B then on B, draw the arcs 1 and 2. Througli 
E and O draw a line and extend it both 
upward and downward until it cuts the 
circle at C and D. 

How do AB and CD comparo in length? 

With ruler and pencil connect Band C; 
C and A ; A and D; D and B. 

The figure BCAD is the desired square. 

exercises 
1. Draw a square inside of a circle of 1" radius; of IJ" 
diameter; of 1|" diameter. 




274 



BATIOKAt OSAITMAB flCHOOl AMTttltmc 



8169. To Model a S-Inch Cabe 

Hall Fiir 1^^ «y 'n««pendeiice 
in i'o**'^*?-^™® <?»y8t«l8 found 

form C^nlr ^'" ^ ^-^^ 

tenors ^^vr'^p--'-' -pat- 

pendacular^ rf^. «« in ProK 

^iRi ' ?; l^-' ,*''■ Problem VI 

p. 181. On this line set off 3-ii,ch 

spaces and complete the deverif 

figure. Leave flaps as inSS 




PlODBB MS 

Old Independence Hall, PMladelphto 



ment, as shown in the tigure. 
Crease the paper along the lines over 
the edge of a ruler and fold up a 3- 

Seluherr^"^'^^^^^^'^*^"^ 

1. What is the area of each sur- 
face? of all the surfaces? 

2. How many edges has the cube? 
How many faxjes meet in each edge? 

The surfaces of the cube meet 
Cat'tU".*'*^'^"^'^™''^"-*. 

op,-, "a ■ P^™^" ^«r(ices (ver'-tl- 

ses). A single corner is a wrL. 

3. How many vertices has the cube? 
How many edges meet at each vertex? 

The edges meet each other in <he 
corners of the cube forming ;o/Sr' 

4. In how many faces does each vertex lie? 

i)oLs?r;oLr?^^"^*^^^^-^^'^^'^^«^-^ 




Figure 144 
Development of 9f cube 



I>o surfaces? 



COKSTHUCTIVE GEOMETfiY 2'J'5 

6. What are the limits of cubes? of surfaces? of lines'? 

Patterns of the surfaces of figures like those of Fig. 144 will 
hereafter be called developments, 

7. Place an inch cube in the corner of the 3-inch cube. IIow 
many inch cubes will fill the row along one edge of the larger 
cube? 

8. How many such rows will form a layer on the bottom? 

9. How many such layers will fill the cube? 

10. How many inch cubes will fill a 3 -inch cube? 

11. How many cubic inches are there in a 4-inch cube? in a 
10-inch cube? in an a;-inch cube? 

§170. To Model a Square Prism. 



.d 



^^^^h 



Draw a straight line ad (Fig. /^ 

1 45) 12 inches long, and make ah and ay 
dc each 2 inches long. Draw en and 
fo perpendicular to ad at b and c. 
Lay off distances eS, hg^ gl^ In^fc^ chj 
km, mo^ each equal to 2 inches. 
Draw ef^jk^ etc., and complete the 
development. Provide flaps; cut 
and paste the model. • figubbhs 

Development of the Square Prtsm 

1. What is the area of each of the end surfaces? 

2. Give the area of each side surface; of all the side surfaces. 

3. How many inch cubes will the model hold? 

4. By measuring the edges how could you find the number of 
cubic inches the model will hold? 

5. How many cubic inches would it hold if it were 1 inch 
longer? twice as long? 

6. If every line in the pattern were twice as long as in the 
development, how would you answer questions 1 to 4? 

Pupils should make their own models, different pupils using 
different lengths of lines and even devising different forms of 
pattern. For example, let one pupil use a 3-inch fundamental 
distance instead of an inch, another a 4-inch distance, and so on, 
each comparing his completed model with other models. 



il6 



HATIOKAL OBAltMAB 80S00L ABITHMETtC 



§171. To Model a Plat Prism. 



>fy 



^ 



k 




I 






f 

1 

! 
; 


h 
n 


c 




d 
P 






r 




s 


!Z 



Figure 146 
Development^of Flat.Prism 



Draw a line ae (Fig. 146) 18 in. 
long and mark off ed = d in., dc = 
6 in., cb= 3 in., and ba = 6 inches. 

At c and d draw perpendiculars 
to ae and extend them 11 in. below 
c and d to r and «, and 3" above c 
and dtok and /. 

Make co and dp 8 in. long and 
draw mq through o and p. 

Complete the development by 
drawing the necessary parallels. 

Provide it with the necessary 
flaps and paste up the model, leaving 
the top ckdl open. 



1. With the aid of the model of an inch cube, find how 
many cubic inches would fill the flat prism. 

2. To what is the product of the 3 different edges equal? 

3. How could you obtain the capacity in cubic inches of 
such a prism by measuring the lengths of its edges? 

4. How many square inches in the whole surface of the flat 
prism of Fig. 146? 



§172. CompariBon of Prisms. 

(a) A right prism. 

Using the lengths of lines as 
shown in Fig. 147, draw a pattern on 
heavy paper and construct the model 
of a right prism as was done in 
Fig. 145. Provide the edges with 
flaps and paste up the model, leav- 
ing one end open, so that it may be 
filled with sand. How many cubic 
inches of sand will just flll the model? 

How can you find the capacity 
of the model by using the lengths 
of the edges? 



s" 3" 
s;^^ ^ 

3" 3" 



FIOUBE 147 
Development of Bfght Prism 



CONSTRUCTIVE GEOMETRY 



277 



(b) An oblique prism with parallelograms for bases. 

Similarly draw a pattern, cut, 
fold, and paste up a model of the 
oblique prism, as shown in Pig. 
148, leaving an end unpasted for 
sand. 

When the model of the right 
prism is just full of sand, the 
sides not being bulged out with e^-?- 
the sand, pour the sand into the 
model of the oblique prism. Does 
it fill the second model? 

What is the ratio of the capac- 
ities of the two models? 

What is the ratio of the areas 
of the bases (ends)? 

(c) A triangular prism. 



X 






X 


a 






d 


/ 


3" 


3" 


\a; 


/S" 


h 


c 


^''X 


/^ X 




N 


r 




3'* 



FiGTTBID 148 

Derelopment of ParaUelogram Prism 



X 




-x 




1^ 




r 








V' 




^ . 


•^ 1'' 








y'/} 


^^v 


V' 




1'' 


// 






















'*^ 













L-i:— : — 




.- ft 





Model a triangular 
prism like the one shown 
in Fig. 149. 

Compare the capacity 
of this model with that 
of the model of a square 
prism l"xl"x4". What 
IS their ratio? 

Model a triangular 

prism such as that of 

Fig. 149, but use for 

lengths 3" instead of 1" 

and 7" instead of 4". 

How does its capacity compare with that of the prism of Fig. 147? 

of Fig. 148? Give the ratio in each case. 

Compare the ratios of the bases. 

§173. Volume of an Oblique Prism. 

The volume of a figure is the number of 
cubical units in the space enclosed by its 
bounding surfaces. 

1. How many cubic inches are there ^^ --:;:^' j 

in a straight pile of visiting cards 2" high, if figubb im 

each card is 2" x 3" ? ^^^ ^S ^'^''' 



Figure 149 
Development of Triangular Prism 



f 




278 RATIOKAL GRAMMAR SCHOOL ARITHMETIC 

2. How many cubic inches would there be in the pile if it 
were 5" high? 9" high? a in. high? 

3. Push the straight pile of problem 1 over as in Fig. 151. 
How many cubic inches of paper are there in this oblique pile? 

4. Has the height of the pile been 
changed in Fig. 151? has the area of 
the base? has the volume? 

5. How can you find the number of 
cubic units in a right prism from tbo 

Figure 151 area of its base and its height? in an 

Oblique Pile, or Oblique Prism oblique prism? 

6. Find the volumes of square prisms having edges of the 
following lengths : 

(1) 3" X 5" X 6" ; (4) 4f ' x 6' x 12' ; (7) a in. x ^ in. x c in. ; 

(2) 6" X 5" X 9" ; (5) 16' x lOf x 20' ; (8) a; f t. x v ft. x ;2; f t. ; 

(3) 15"x9"x8"; (6) 45" x 16|" x 21"; i9)myd.xnyd.xpjd. 

§174. Paper-Folding. 

For the following exercises in paper-folding any moderately 
thick, glazed paper will do. Tinted or colored paper, without 
lines, will however show the creases more clearly. It is con- 
venient to have the paper cut into pieces, about 4" square. Such 
paper is inexpensive and may be had of any stationery dealer. 

Problem I. — At a chosen point on a line, make a perpendicular 
to the line, by folding paper. 

Explanation. — Fold one part of a piece of paper over upon the other 
and crease the paper along the fold, as at AB, by drawing the finger along 
the fold. Taking D to denote the chosen point, fold the paper over the 
point 2), and bring the two parts, DA and DB, of the crease AB 
exactly together. Hold the paper firmly in this 
position and crease the paper along the line DC. 

Compare the portion of the paper between the 
creases DB and DC with the portion between the 
creases DA and DC. How do the angles BDC and 
CD A compare in size? 

Definitions. — When two lines meet in this way 
making the angles at their point of meeting (inter- 
section) equal, the lines are said to be perpendicular 
to each other, and each is called a perpendicular to 
the other. 

The angles thus formed are called right angles. Figubb 152 




COKSTRUCTIVE GEOMETRY 



279 



Problem II. — Bisect an angle, by folding paper. 




Figure 153 



Explanation.— Crease two lines, as OA and OB, 
lying across each other. They make the angle 
AOB. 

Now fold the paper over, and bring the crease 
OA down on OB. Holding the creases firmly to- 
gether crease the bisector OC. 

How do the angles AOC and BOC compare in 
size? Do they fit? 

What is the ratio of angle AOC to angle BOC? 
of AOB to AOC? 

How could the angle AOB be divided by creases 
into 4 equal parts? 



Problem III. — Crease three non-parallel lines. 

Explanation. — Crease AB in any position. Crease CD in any posi- 
tion, not parallel to AB. Finally, crease EF in any position not parallel 
to either AB or CD, Fig. 154. 

In general, in how many 
points do three non-parallel 
lines cross each other? 

Can you crease three lines, 
no two of which are parallel, 
in such a way as to obtain just 
two crossing points (intersec- 
tions)? Try it. 

Crease three non-parallel 
lines in such positions as to 
give but one intersection (see 
Fig. 155). Lines which go 
through the same point are 
called concuTrrent lines. 



k ^ 


—!- 


r 


B 
l> 





iC F, 


A 


O B 


E: 


\D .1 



FIOUBE 154 



FIGUBB 155 



Problem IV. — Crease the bisectors of the 3 angles of a 
triangle. 



Explanation. — Crease out a triangle such as 
ABC. 

Then crease the bisector of each angle, as in 
Problem II. Work carefully. 

1. How do the bisectors cross each other? 

2. Since the three bisectors of the three 
angles of a triangle all go through 0, what 
name would be applied to them 




Figure 150 



280 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

Problem V. — Bisect a given line, by paper-folding. 



j mmf r r- .^ |™- — , EXPLANATION. — Crease a line and stick the point 

■ "" ' *^ of a pin through the crease at A and at B, 

AB is the line to be bisected. 

^1 „| Fold the paper over and bring the pin hole at 

*.4^-»™--H.i; ,p-n.^™™nrrt3 B down on the pin hole at A. Press the paper 
down and crease a line across AB, as at C. CD is 

I the perpendicular bisector of AB. 

i ,„, lln „,„ niii J Crease a line from D to B and another from D to 

L - ^-" "^^^ ^ ^^"^ ' ^ '^^ A, When the paper is folded over the line CD, how 

•r^ — do these two creases seem to lie? Ck)mpare the 

FiQUBB IW lengths of DB and of DA. 

1. Crease the perpendicular bisectors of the 3 sides of a triangle 
and find how they cross each other. 

2. What kind of lines are the perpendicular bisectors of the 
sides of a triangle? 

Problem VI.— Crease a square and its diagonals. 



Explanation.— Crease 2 lines, as AB and AX, ' -X":' T 

perpendicular to each other. (See Problem L) 

Fold the paper over the point B and when the ^ ^ 

two parts of the crease AB fit, crease the line BY. q 

Now fold the paper over so that crease BY 
comes down along crease BA, and crease the line 
BC. _^ 

Fold the paper over a line through C, bringing -V, ^' \ 

CX down along CA, and crease CD. * — -^ L — J 

ABCD is the required square. 

FIGURE 168 

1. When the square is folded over the diagonal BC, where 
does D fall? Along what crease does BD lie? CD? 

2. How does the diagonal divide the area of the square? 

3. Fold over and crease the diagonal AD? How does it divide 
the square? 

4. Compare OB with DC; OD with OA. 

5. How do the diagonals of a square divide each other? 
G. How do the diagonals divide the area of the square? 

7. How does AD divide the angle BAC? the angle BDC? 

8. Fold a second square and crease its diagonals (Fig. 159). 
Fold over 0, bringing D down on HO and crease JIG. Similarly 
crease IJF. 



OONSTBUCTIVB GEOMETRY 



281 



r - 


■' 




"^ 


r 


,f 


D 




£ 


o 


!• 




J 


' 1 


i 


t 


1 




9. When the paper is folded over OF along what line does 
ODtaM? OH? 

10. How does OB compare with OD in 
length? How then do .4Z> :md ^(7 compare? 

11. How do HF and HG divide the 
square? 

12. Crease the following lines : GF^ FH^ 
HE^ and EG. How does the area of the 
square GFHE compare with that of ABDC? figurb 159 

Problem VII. — Crease the three perpendiculars from the 
vertices of a triangle to the opposite sides. 

Explanation.-— Crease the triangle ABC, Fig. 
160. Fold the paper over the vertex C, and bring 
the crease DB down along DA, and crease the per- 
pendicular CD. 

Similarly crease AE and BF, Work carefully. 
How do the creases CD, AE and BF cross each 
other? 

What kind of lines are the three ^rpen- 
diculars from the vertices to the opposite sides of 
a triangle? 

Figure ifto 

Problem VIII. — Crease the three perpen- 
dicular bisectors of the sides of a triangle, 
Fig. 161. 

How do they cross? What kind of lines 
are they? 

Problem IX. — Crease a rectangle and its 

diagonals. 

Explanation.— Crease AB, Fig. 162, making the 
distance from A to B two inches. By Problem I, 
crease the perpendiculars AD and BC. Make AD 
and BC each 1" long and crease CD. 

Crease the diagonals, one through A and C, and 
the other through B and D. Call their crossing 
point O. 

Fold the paper over the perpendicular FE, bring- 
ing B down on A. Where does C fall? What 
other line equals CB? OB? OC? 

Fold the paper over the point O so that B falls 
.^ on C, and crea.se the perpendicular HG. What 

FiGTOB 168 other line equals AB? OG? OD? 

How does the intersection, O, of the diagonals divide the diagonals? 




FIGUBB 161 



■J . 


'■ s.^ 




f^;i|w|| 



282 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



§175. Perimeters. — The perimeter of any figure is the sum of the 
lines bounding the figure. Thus, in form (0), Fig. 163, if ^ de- 
note the perimeter, 

(I) p = x + t/ + z. 

1. What does x ■h2x + 4:X mean? IIow may it be more briefly 
written? 

2. What is the coefiicient of x in the answer to problem 1? 

3. Write the perimeter, p, of (1), Fig. 163, in two ways. 

4. Write the perimeter, p, of (13) in three ways. 

5. Write the perimeter, p^ of (12) in three ways. 

6. Write i of the perimeter, p^ of (13) in two ways. 

7. Write an expression showing that the two answers to 6 are 
equal. 




Figure 163 

8. In (15), if a; = 80 rods and y = 40 rods, how many feet in 
the perimeter? 

9. In (1) and (2), if x and y each = 20 rods and one side of 
the triangle rests on the square, making a new figure, omitting 
the common line, what is the perimeter of this new form, in feet? 
How many sides has the new figure thus formed? 

10. Forms (1) and (13) are combined into a single figure. 
Write p for the new figure in two ways, supposing x the same in 
both. 



CONSTRUCTIVE GEOMETRY 283 

§176. Quadrilaterals. 

1. Forms (2), (7), (8), (10), (11), (12), and (13), Fig. 163, aro 
differBnt kinds of quadrilaterals. What is a quadrilateral 9 

2. What quadrilaterals have their opposite sides parallel? 
These figures are parallelograms. Define a parallelogram, 

3. What parallelograms have all their sides equal? What is a 
rhombusf 

4. What rhombus has all its angles equal? What is a square? 

5. What quadrilaterals have their opposite sides equal but con- 
secutive angles not equal? Define a rhomboid. 

6. What parallelograms have their angles all equal? Define a 
rectangle. 

7. What quadrilateral has only one pair of sides parallel? De- 
fine a trapezoid. 

8. Is a trapezoid a parallelogram? Is it a quadrilateral? 

9. Is a rectangle necessarily a quadrilateral? Is it a parallel- 
ogram? a square? May a rectangle be a square? 

10. Is a square necessarily a quadrilateral? Is it a parallel- 
ogram? a rectangle? a rhombus? 

§177. Perimeters of Miscellaneous Figures. 

1. Denote the perimeter of each of the forms in Fig. 163 by 
p and write an equation like (I) in §175, showing the value of p 
for each figure. 

2. Omitting the lines on which the forms join, write an equa- 
tion showing the value oip when (2) and (6), Fig. 163, are 
joined on y, which has the same value in both. 

3. In the same way join (3) and (10) on a, which has the same 
value in each, and write an equation showing the value of p. 
What is the name of the figure thus formed? 

4. Join (2) and (7) in which x and y are equal. If the perimeter 
of the figure thus formed is 240 rods, find the value of x in feet. 

5. What is the length of the perimeter of a figure like (6), Fig. 
163, whose sides are 2 in., a in., and h in. long? whose sides are 
X in., 8 in., and z in. long? like (10), whose sides are c ft., 2c ft., 
V ft., and X ft.? like (12), whose sides aro 2a; rd., 4y rd., 2a: rd., 
and 4^ rd. long? 



284 



BATIONAL GRAMMAR SCHOOL ARITHMETIC 



Write the perimeter (j^) of each of the figures below. 



(i) 

Square 



Rectangle 




b 


c 

« (B) 







y ^ 



y 
y 




(«) 





-x * 





„ 


-b— 


^ 


t 




i 


a? 




a; 


X 




(7) 




i 

i 


1/ 




y 



^ 


—y— 


^ 


h 




6 


a 


(«) 


a 





















X 

y 




y 






(9) 







d d 






UO) 





FiGUBB 164 

In such forms as' those from (o) to (10), Fig. 164, some line, 
or lines, must be found by subtracting others. In (6) for example, 
note that the ends are each x — y. The perimeter is then a? + a; + 
(x — y) •\-y -\-y + {x- y) = 4a; — y — y+ y + y = 4a: — 2y+ 2y=4a?. 

All sides not lettered must be expressed without using other 
letters than those given on the figure. The perimeter means the 
sum of all the lines that bound the strip, or surface, of the 
figure. 

Note. — One-half the difference of a line m and a line n is written 
—^ — or! {yn — n), 

L In (8), a = 15', h = 12', x = 25' and v = 30'. Find the length 
of the perimeter of the figure. 

2. Find the area of (8) enclosed by the solid lines. 

3. Make and solve other similar problems. 




CONSTRUCTIVE GEOHETBY 285 

§178. Heasnring Angles and Arcs. 

ORAL WORK 

We may measure the amount of turning of each clock hand in 
either of two ways : 

(1) By the length of the circular arc 
passed over by the tip of the rotating 
hand; 

(2) By the wedge-shaped space having 
its point at the hand-post, A, over 
which the stem, A-III, A-VI, etc., of 
the rotating hand moves. 

While the clock hand turns from XII 
to III its tip moves over 1 quadrant of arc 
FiGURB 165 ^^^ j^g ^^^^ moves over a right angle. 

1. While the hand turns from XII to VI over how many 
quadrants does its tip move? Over how many right angles does its 
stem turn? 

2. Answer same questions for a turn of the hand from XII to 
IX ; from XII to XII again ; from XII around through XII to III. 

3. Over what part of a right angle does the stem of the hand 
move while the hand is passing from XII to I? from XII to II? 
from III to IV? from VI to VIII? 

4. Over what part of a quadrant does the tip of the hand pass 
in each case of problem 3? 

5. Over what part of a right angle does the stem of the hand 

move while the tip moves 1 min. along the arc? In the same 

case over what part of a quadrant does the tip move? 

Definitions. — Any wedge-shaped part of the face moved over by 
the stem of a clock hand as it turns around the hand -post is called an 
angle. The curve passed over by the tip of the hand, while the stem of 
the hand moves over the angle, is called the arc of the angle. 

Illustrations. — The wedge-shaped spaces, having their points at A, 
and included between any two positions of the hand, as A-XII and A-I, 
A-1 and A-III, A-I and A-VI, are all angles; the space swept over by the 
hand, A-I, as it moves around through II, III, IV, V, VI, etc., to IX is 
also an angle. 

6. As the hand moves from A-XII to A-VI, that is, so that the 
two positions of the hand are in the same straight line, the hand 
moves over a straight angle, A straight angle equals how many 
right angles? 



280 



RATIOKAL GRAMMAR SCHOOL ARITHMETIC 




7. The arc of a straight angle equals how many quadrants? 
How many quadrants make a complete circle? 

8. How many 5-min. spaces make the circumference of a 
complete circle? How many 1-min. spaces make the circumfer- 
ence of a complete circle? 

9. What part of a right angle is passed over by the stem of the 
hand as its tip moves from one end to the other of a 1-min. space? 

10. If lines were drawn from the hand-post to the ends of all 
the 1-min. spaces, the whole face of the clock would be divided 

up into how many small 
angles? 

The instrument in 
common use for meas- 
uring angles and arcs 
is the protractor. (See 
Fig. IGG.) 

11. Instead of divid- 
ing the right angle up 
by radiating lines into 
15 equal parts, the protractor divides the right angle into 90 equal 
parts, one of which is called the angular degree. These same lines 
would divide the quadrant up into how many equal parts? Each 
of these parts is a degree of arc. 

12. How many angular degrees are there in a straight angle? 
How many degrees of arc in the arc of a straight angle? 

13. How many angular degrees are swept over by a clock hand 
while moving entirely around once? How many degrees of arc in 
the circumference of a circle? 

14. How many degrees of angle are passed over by the stem of 
the minute hand in two hours? In the same time how many 
degrees of arc are passed over by the tip of the hand? 

15. A sextant is ^ of a circumference; how many degrees of 
arc are there in a sextant? 

16. An octant is ^ of a circumference; how many degrees 
of arc are there in an octant? 

17. Study the protractor, Figs. 166 and 167; notice how its 
marks are numbered. How many degrees of arc are there between 



Figure 166 



COlfSTRUCTIVE GEOMETRY 



287 



the closest lines on the outer edge? How many degrees of angle 
are there between the lines which converge toward the center? 

18. For smaller angles a shorter unit is 75V <^^ ^ degree of angle 
and the smaller unit is called the minute of angle. Vir ^^ ^^^ 




Figure 167 



minute, called the second^ is a still smaller unit. How many 
minutes of angle in a right angle? in a straight angle? in a com- 
plete revolution, or perigon? How many seconds, in each case? 

19. The arcs between the sides of the minute and the second 
angles are the minute and the second of arc. How many minutes 
of arc in a quadrant, in a sextant, in an octant, in a circumfer- 
ence? How many seconds, in each case? 

A paper protractor can be purchased at any bookstore, and 
each pupil should supply himself with one. 

To measure an angle the center of the protractor is placed on 
the vertex A, and the line through A is placed along one side 
AD (see Fig. 168). The reading of the mark below which the other 




FlGUBE 168 



side, AF, falls is the number of degrees in the angle, or in the 
arc of the protractor included between AD and AF. 




288 EATIONAL GRAMMAR SCHOOL ARITHMETIO 

WRITTEN WORK 

20. Draw a triangle, and, with a protractor, measure its angle*. 
To what is the sum of all three equal? 

21. With ruler draw a half a dozen triangles of different shapes, 
carefully measure each angle, and find the sum of tlie three 
for each triangle. 

22. Measure with the protractor the number of degrees in the 
angle at one comer of the page of this book. 

23. Draw two straight lines crossing each other as in Fig. 160. 
With the protractor measure and compare a and c\ b 
and d» 

The angles a and c, or b and rf, lying opposite 
each other, are called opposite angles^ or vertical 
FiGUBB 169 atigles. 

24. Draw two crossing lines in different positions, and in each 
case compare the measures of a pair of opposite angles. What 
do you find to be true? 

25. Draw a pair of parallel lines, as a and J, Fig. 170, 
and draw a third straight line c cutting across the 
parallels. Measure with a protractor and compare the •- 
four angles, in which 1 is written in Fig. 170. Measure 
and compare those in which 2 is written. What do 

. _ f , , Q FlGUBB 170 

you find to be true? 

26. Draw a line perpendicular to another, see §168, and measure 

all the angles formed. What do you find?* 

Definitions. — ^The lines which include the angle ^^yq sides of the angle. 
The point where the sides meet is the vertex of the angle. 

27. Draw a quadrilateral of any irregular shape and measuro 
each of its 4 angles. To what is the sum equal? Try another 
quadrilateral and see whether you obtain the same sum. 

28. Draw a given angle at a point on a given line. 

Explanation.— Let the given line be ED, the given angle 35", and the 
given point A, Fig. 168. Place the protractor with its center at A, and 
with the diameter of the protractor along the line ED. Make a dot 
opposite the 35" mark on the protractor. With the ruler draw a straight 
line through A and this dot to F. The angle DAF is the required angle. 

29. Draw a straight line, mark a point on the line and draw 
the angle 76°; 95°; 100°; 135°; 55°30'. 




ifijEr 



COK8TR0CTI7'E aEOMETRY 



289 



Table op Units of Angle and Arc Measurement 
60 seconds (") = 1 minute (') 
60 minutes = 1 degree (**) 
860 degrees = 1 circumference (or perigon) 
90 degrees = 1 quadrant (or right angle) 

Table op Equivalents 
1,296,000" 
21,600' 
360** 
4 quadrants 



= 1 circle 



§179. The Sum and Difference of Angles. 

Exercise 1. — With ruler and compasses, draw AB and CD 
perpendicular to each other. 

1. Suppose a protractor supplied 
with a pointer as shown in Fig. 171, 
the center of the protractor being 
placed at the point, 0, over how many 
degrees of arc would the tip of the 
pointer turn while the stem of the 
hand turns from line OB to line 0(7? 
from OB through OCU) OAl 

2. How many degrees are there in 
the angle jSOCT in angle BOAl in 
COD'i in A on? in the angle from OA 
around through OD to OB? from OA 
around to 00? from OA entirely 
around to OA again? 

3. How many degrees of angle fill the space around a point, 
as 0, on one side of a straight line, 
as AB? on both sides? 



Definitions. — An angle that Is smaller 
than a right angle is called an acute angle 
(see Fig. 172). An angle that is larger than 
a right angle is called an obtuse angle. 





Acute Angles Obtuse Angles 
FXOURB 173 



200 



RATIONAL OHAMMAK SCHOOL ARITHMETIO 




Figure 173 



Exercise 2. — Find the sum of two angles 

1. Draw two acute angles like those at the top of Fig. 173,care- 
j^ f iilly cut them out, and place them as 

^^'^ in the lower part of the figure. Posh 

their vertices and nearer sides en- 
tirely together. Draw two lines along 
the other sides of the angle. Bemoye 
the angles and have an angle like that 
on the left, which is the sum of the 
two angles. 

2. Calling the larger angle re, and the smaller y, what denote 
the last angle? 

3. Draw an angle equal to the sum of an acute and an obtnae 
angle. 

Exercise 3. — Find the difference of two angles. 

1. Draw two angles like those at 
the top of Fig. 174, cut them out, and 
place them as in the lower, left-hand 
part of the figure. Mark along the 
lower side of the upper angle and 
cut along the mark. The lower part 
of the larger angle (shown on the 
right) equals the difference of the two 
angles. 




FIGDRB 174 





Figure 175 



Figure 176 



2. If the larger angle equals x degrees and the smaller equals 
y degrees, how many degrees are there in the difference? 



CONSTECCTIVE 6E0METBT 



291 




Figure 177 



Definitions. — ^Two angles whose sum equals 
a right angle, or 90°, are called complemental 
angles (Fig. 175). Two angles whose sum 
equals 2 right angles, or 180'', are called sup- 
plemental angles (Fig. 176). 

The sum of all the angles that just cover 
the plane on one side of a straight line is equal 
to how many right angles (Fig. 177)? To how 
many degrees is this sum equal? 

Exercise 4. — Draw an equilateral triangle (Problem VII, p. 
102). Tear off a corner and fit it over each of the other corners in 
turn. How do the three angles compare in size. Tear off the 
other corners and place them as in Fig. 178. To what is the sum 
of all three angles equal? 

If one of the angles is x degrees, how many degrees are there 
in the sum of all three angles? As the sum of the three angles 





rs>^ 



Figure 179 



Figure 178 

equals both dx degrees and also 180°, we may write the equation 

3x = 180. 
If Sx = 180, to what is x equal? How many degrees are there 
in one of the angles of an equilateral triangle? 

Exercise 5. — Draw any scalene triangle. Tear off the corners 
and place them as in Fig. 179. To how many right angles is the 
sum of all the angles of the triangle equal? To how many 
degrees? 

1. Letting a?, y, and z denote the numbers of degrees in the 
respective angles, in what other way may we write the sum of 
all three? What equation piay we then write? Tell what the 
equation x-{-y + z = 1S0° means. 

2. Draw other scalene triangles, tear off the corners, place them 
as in the figure, and find whether x + y + z = 180° in all cases. 



202 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

3, Crease a right triangle and find whether the equation, 
x-{-y + z = 180°, is true for it. 

4. Cut along the creases and tear off the two acute angles of a 
carefully creased right triangle and fit them over a carefully creased 
right angle? What seems to be true? If this is true what equation 
may you write for the sum of the two acute angles {x and y) of a 
right triangle? 

Exercise 6. — Draw a quadrilateral and cut it along a straight 
line from one corner to the opposite comer as in Fig. 180. Such 
a line as is indicated by the cut is called a diagonal. 

1. The cut divides the quadrilateral 
^"^-*--.. — into figures of what shape? 
J~^y^ ^^.^"''^ ^- To what is the sum of the three 
ly* e/ l^__«_2/ angles of each part equal? 

FIGURE 180 3. To what is the sum of all six 

angles of the two triangles equal? 

4. To what is the sum of all 4 angles of the quadrilateral equal? 

5. Do your answers hold true for the parallelogram of Fig. 180? 

6. Do they hoi d good for any shape of quadrilateral you can draw? 

7. To what then is the sum of the four angles, x^ y, z^ and w, 
of any quadrilateral equal? 

Exercise 7. — Draw a hexagon and cut it along diagonals as 
•shown in Fig. 181. . 

1. Into figures of what shape 
is the hexagon divided? 

2. Into how many such figures 
do the diagonal cuts divide the 
hexagon? 

3. To how many degrees is 
the sum of all the angles of all the triangles equal? 

4. To how many right angles is the sum of all the angles of 
the hexagon equal? 

5. Do all your answers hold true for the regular hexagon 
(sides equal and angles equal) of Fig. 181 also? 

6. In each of these hexagons how many less triangles than 
sides of the uncut hexagons are there? 




Figure 181 




CONSTBUOTIVE GEOMETRY 293 

7. Draw an S-sided figure and find how many triangles the 
diagonal cuts from any vertex would give. 

8. How many less triangles than sides wei:e given by the 
quadrilaterals of Fig. 180? 

9. To how many right angles is the sum of all the angles of 
any 15-sided figure equal? of an w-sided figure? 

10. The angles of a regular hexagon are all equal. How many 
degrees does each contain? 

Exercise 8.— Draw a rectangle and cut it out. Cut it along 
a diagonal, and denote the angles by letters, as shown in Fig. 182. 

1. Turn the lower right-hand 
triangle around and place it on the 
upper piece so that e may fall at a, 
/ at c^ and d at J. Can you make 
the parts fit? 

2. How does a diagonal divide a rectangle? 

3. Show the relative length of sides m and n by an equation ; 
of h and l\ of the angles a and e\ c and /; b and d; b + e and 
a + d. 

4. From these equations point out those which show that the 
opposite sides of a rectangle are equal. 

5. Show from your equations how the opposite angles compare 
in size. 

6. With parallel rulers draw a parallelogram and cut it as sug- 
gested by Fig. .182. Answer questions 1-5 for the parallelogram. 

7. Write the perimeters of the parallelogram and of the rect- 
angle of Figs. 180 and 182. 

§180. Products of Sums and Differences of Lines. 

1. What is the area of A? of B? of C? 

Note. — The product of a and x-^y is writ- 
ten a(X'\-y) and is read **a times the sum 

^- + 2/." 

^ •*■ y 2. Read the equation ax + ay ^a (x + t/) 

FiGURai83 ^^^ explain its meanmg from Fig. 183. 

dp Draw a figure and show that 

axi-ay + az^a (a; -f y + 2;). 





X 




« 


a 


A 


B I 










Q 


c 1 



294 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



written 



4. What is the area of each of the parts of Fig. 184? 
Note, a-^b times x-^y is 

I fp] 5. From Fig. 184 show the meaning of 

^ each product in the equ ation {a + b) {x + y) 

= ax-j-ay + bx + by. Also show from Fig. 



x + y 



FIGUBB 18i 



184 why the equation is true. 

6. What is the area of each part of Fig. 185? 

7. Write an equation showing how to mul- 
tiply x + y hj itself, or how to square x + y. 

8. Point out from Fig. 185 the meaning of 

{x + yy=^a^ + 2xy + y^. 

9. Point out from Fig. 186 the meaning of 

b {a — c) =ab -'be. 



x+y 



[SSI 



X y 
FiGUBS 186 







t 








t 


c ' 




o 
1 








i 






C3 




b 

PiGi: 


fBB] 


b 

L86 


o+b 

FIGUI 


IE 18 


- a -• 
7 





a -> 

::::io 



10. Draw a figure and show that 

b {a + cl — c) =ab + bd — be. 
11, Show from Fig. 187 that 

{a + b){a-b)=a^-h'. 

-^-^ 12. Show from Fig. 188 that 

Figure 188 (a -b) (a-b) =a^ — 2ab + J*. 

13. What is the area of the rectangle A BCD (Fig. 189)? of 5? 

14. What is the length of the dotted 
part of BC? 

15. What is the area of B? 
IG. What is the area of the cross-ruled 

part? 

17. Write the areas of the cross-lined 
figures below (Fig. 190). 




Figure 189 



COKSTRUCTIVE CJEOMETRY 



295 



Call the difference of the two areas d in each case, and answei 
with an equation. 

il) (») (3) (4) 




FiGUBS 190 

18. A lot, 50' X 176', JB occupied by a house' which, with its 
porch, covers a 40' square. A cement walk 3^ wide runri from 
the front line of the lot to the porch, and from the back door of 
the house to the rear line of the lot. The rest of the lot is covered 
with grass. How many square feet of grass are there? 

19. A man cuts an 18" strip of grass entirely around a rectan- 
gular lawn, 75' x 175' ; how many square feet of grass does he cut? 

20. He then cuts another 18" strip around just inside of the 
strip mentioned in problem 19; how many square feet of grass 
does he cut the second time round? 

21. A farmer reaps a 12' swath entirely round a rectangular 
wheat-field that is 40 rd. x 80 rd. How many square rods does 
he reap the first time round. (12' = y\ rd.) 

22. He again reaps a 12' swath entirely round the field. How 
many square rods does he reap the second round? the third round? 

23. A farmer plows an 18" strip entirely round a rectangular 
piece of ground 20 rd. x 80 rd. How many sq. ft. does he plow? 

24. He continues plowing round the rectangle (prob. 23) 
until he has a strip plowed 1 rd. wide entirely round it. How 
many acres has he then plowed? 

25. How many acres has he plowed when the plowed strip is 
2 rd. wide? 4 rd. wide? 

26. How many acres has he plowed when the unplowed strip 
of prob. 23 is 4 rd. wide, the plowed strip being of the same width 
all round the rectangle? 

27. How many acres have been plowed when the unplowed 
strip remaining is 3^ rd. wide? 2^ rd. wide? 



296 



BAtlONAL QEAMMAft SCHOOL ARlTHMEtlO 




PLACES ON THE EARTH 297 

§181. Locating and Describing Places on the Earth. 

Fig. 191 is a map of the eastern and the western hemispheres 
of the earth,, showing the lines that are used for locating and 
describing places on the earth. An imaginary line, called the 
equator^ runs entirely round the earth, dividing its surface into 
the northern and the southern hemispheres. This equator is 
shown on the map by the heavy horizontal line. 

The heavy curved line going through the north and south 
poles of the eastern half of the map and crossing the equator at 
0° is called the^rime meridian, 

ORAL WORK 

1. What do the numbers 15°, 30°, 45°, etc., mean, that stand 
along the equator to the right and left of 0°? 

2. What part of the map is north? south? east? west? 

3. How many degrees are there between the curves that run 
up and down on the map through the poles? 

4. How many degrees are there between the curves that run 
across the map from right to left? 

5. Show the place on the map that is 15° (meaning 15 degrees) 
west of the zero (0°) point; 30° (30 degrees) west of 0° ; 60° W. ; 
105° W. ; 150° W. 

6. Show the place that is 15° east of the 0° point; 45° E. ; 
60° E.; 90° E.; 120° E.; 150" E. ; 165° E. 

7. Point to the place that is 15° north of 0°; 30° N.; 30° 
south; 45° S.; 60° N. ; 75° S.;.90° N. ; 90° S. 

8. Show the place that is 30° W. and 15° X. ; 30° W. and 30° 
N.; 30° W. and 15° S.; 60° W. and 30° S.; 60° E. and 15° N.; 
60° E. and 30° S. 

9. Point out the places that have the following positions: 

(1) 30° W. and 75° N.; (8) 105° W. and 30° N.; 

(2) 30° W. and 30° S. ; (9) 60° W. and 60" S. ; 

(3) 45° E. and 15° S. ; (10) 105° W. and 45° S. ; 

(4) 60° E. and 45° N. ; (11) 90° E. and 30° N. ; 

(5) 45° W. and 45° N. ; (12) 90° E. and 90° N. ; 

(6) 45° W. and 45° S. ; (13) 180° W. and 75° S.; 

(7) 45° E. and 45° S. ; (14) 90° W. and 90° N. 



298 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

Degrees measured toward the west and toward the north are 
written with a plus (+) sign hefore them. Degrees measured 
toward the east and toward the south are nmrked with a minus 
(— ) sign before them. 

10. Point out the following places : 



(1) +15°, +30° 

(2) +15°, -30° 

(3) +60°, -75° 

(4) -60°, -75° 

(5) -75°, +60° 

(6) -75°, -60° 



(7) +100°, -45° 

(8) +100°, +75° 

(9) +160°, +80° 

(10) - 40°, +65° 

(11) -140°, -35° 

(12) -120°, -65° 



WRITTEN WORK 

1. Write down about the correct distances in degrees westward 
or eastward, and northward or southward, of the mouth of the 
Amazon Eiver; of the Mississippi River; of the Nile River; of 
the Ohio River; of the Niger River (Africa). 

2. Write about the positions of these places, using + or — 
signs: Washington, D. C. ; New York City; Boston; Chicago; 
Denver; New Orleans; London; Paris; Berlin; the south point 
of Florida; the south point of South America; the east point of 
South America. 

3. About how many degrees long is South America? Africa? 
The Nile River? 

4. About how many degrees wide is South America? Africa? 
The United States? 

The curved lines running from right to left are called parallels; 
those running from north to south are meridians, 

5. About how many degrees wide is the United States along 
the 30° parallel? along the 45° parallel? 

6. About how many degrees wide is South America along the 
equator? along the -30° parallel? the -15° parallel? 

7. Answer similar questions for Africa. 

8. How many degrees are there between the prime meridian 
and the meridian of the mouth of the Nile River? between the 
prime meridian and the meridian of the southeast point of Arabia? 
of the southwest point of Arabia? 



PLACES ON THE EARTH 



299 



9. How many degrees are there between the 45th meridian and 
the meridian of the mouth of the Mississippi Eiver? 

10. Chicago is about how many degrees west of New York City, 
that is, about how many degrees are there between a meridian 
through New York City and a meridian through Chicago? 

11. About how many degrees is Chicago north of New Orleans? 
of Panama? of Rio de Janeiro? of the mouth of the Amazon River? 

12. About how many degrees west from New York City is 
Denver? San Francisco? The mouth of the Ohio River? 

Distances measured westward or eastward in degrees along a 
parallel from the prime meridian are called longitudes. Distances 
measured in degrees along a meridian northward or southward 
are called latitudes, 

13. About what is the difference between the longitude of 
New York and of Chicago? Boston and Denver? Philadelphia 
and New Orleans? New Orleans and San Francisco? Washington 
and Savannah? 

14. Give the differences of latitude of the same pairs of places. 
The latitude and longitude in degrees of some important sea- 
ports are given here : 



PliACB LATITITDB LONGITITDB 

Cape Town . . . —33.93 — 18.47 

Hamburg .... +53.55 — 9.97 

Honolulu +21.30 +157.85 

Manila +14.67 —121.00 

Panama + 9.00 + 79.52 

Quebec +46.80 + 71.32 



Latitttdb Longitxtde 

. +51.85 + 8.28 



Place 
Queenstown 

San Francisco. +37.78 +122.43 

Savannah +32.02 + 81.12 

Stockholm .... +59.38 + 18.05 

Valparaiso —33.00 + 71.63 

Yokohama .... +35.50 —139.58 



15. What is the difference of latitude of San Francisco and 
Honolulu? The difference of longitude? 

16. What are the differences of latitude and of longitude of 
San Francisco and Yokohama? of San Francisco and Valparaiso? 
of Hamburg and Savannah? of Queenstown and Quebec? of Cape 
Town and Stockholm? of Panama and Manila? of Panama and 
Valparaiso? of Manila and Yokohama? of Quebec and Panama? 

17. Other problems may be made from the table. 

18. Read the differences of latitude and of longitude of any 
places given on maps in your geographies. 



300 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

§182. Longitude and Time. 

The earth may be regarded as an immense sphere turning, like 
a top, round one of its diameters as an axis from west to east 
once every 24 hr. This carries the surface of the earth and 
all objects fixed upon it (as a schoolhouse) round through 360"" 
in 24 hours. 

Measuring one complete turn (rotation) of the earth in degrees, 
it may be said to be equivalent to 360°; measuring it in time, it 
may be said to be equivalent to 24 hours. 

This gives us the following tables of equivalent measures: 

360° correspond to 24 hr. ; 

1° corresponds to ^^^ of 24 hr. = -^^ hr. = 4 min. of time; 
1' corresponds to -^ ot 4 min. = ^^ min. = 4 sec. of time ; 
1" corresponds to -^^ of 4 sec. = j^ s®^- ^^ time. 

24 hr. correspond to 360° ; 
1 hr. corresponds to 15° ; 

1 min. of time corresponds to -^ of 15° = J** = 15'; 
1 sec. of time corresponds to ^ of 15' = i' = 15". 

All objects seen on the sky, as the' sun, may be regarded as 
stationary, while the turning of the earth carries us past them 
from the west toward the east. This makes the sun appear to 
rise in the east, move over, and set in the west. 

1. Will the sun pass over eastern or western places earlier? 
Which places have later local* times, those over which the sun 
passes earlier, or later? Which places have earlier local times, 
eastern places, or western places? 

2. What time is it at a place 45° west of Washington when 

it is 10 o'clock at Washington? At the same instant, what time 

is it at a place 30° east of Washington? 

Definition. — Longitude is the distance in degrees, minutes and 
seconds (of arc) due eastward or westward from a chosen meridian, called 
the prime meridian. Astronomers and navigators have agreed that the 
prime meridian shall be the meridian of the Royal Observatory at Green- 
wich, England. 

3. The difference between the local times of two places is 4hr. 
3 min. ; what is the difference in longitude between them? 

* Local sun time Is obtained by setting timepieces at XII as tbe sun crosses the 
meridian. 



LONGITUDE AND TIME 301 

4. The difference of longitude between two places is 105° 45'; 
what is the difference of their local times? 

5. It is 4: 50 (4 hr. 50 min.) p.m. at a certain place and 1 : 48 
p.m. at another; which place is east of the other and what is the 
difference of their longitudes? 

6. Two men met in Chicago with their watches keeping the 
correct local times of the places whence they came. On comparing 
their times one watch showed 10:47 p.m., and the other 3:18 
p.m., when it was 9: 26 p.m. in Chicago. From which direction 
did each man come? 

7. If the watches (problem 6) were keeping the correct local 
times of their places, what were the differences of longitude 
between the places and Chicago? 

8. Explain how it happened that the announcement of Queen 
Victoria's death was read in the Chicago dailies at an earlier hour 
than that borne by the announcement itself. 

9. How may it happen that a cable message sent Wednesday 
forenoon may be received at a remote place Tuesday? 

10. The local times of two ships at sea differ by 4 hr. 18 min. 
15.6 sec. ; what is the difference of their longitudes? 

11. One ship is in longitude 186° 40' 12" west and another is 
in longitude 20° 16' 48" west; what is the difference of their longi- 
tudes? of their times? 

12. The longitude of one ship is 3° 28' 10" west and that of 
another is 18' 38" east. What is the difference of their longitudes? 
of their times? 

13. The longitude of the Observatory of Madrid, Spain, is 
3° 41' 17" west and that of the Berlin Observatory is 13° 23' 43" 
east. What is the difference of their longitudes? of their 
times? 

14. The longitudes of the Cambridge (Eng.) and of the Paris 
Observatories are 5' 41.25" east and 2° 21' 14.55" east respec- 
tively. What are the differences of their longitudes? of their 
times? 

15. When the sun is on your meridian, that is, when it is noon 
at your place, at what places on the earth is it forenoon? after- 
noon? night? midnight? 



302 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

The longitudes from Greenwich, Eng., of places are also 
often given (as below) in hours, minutes and seconds of time. 
The plus (+) sign means that the place is west of the meridian 
of Greenwich, and the minus (— ), that the place is east of this 
meridian. 

j^ Ar™ liONGITITDB TBOM «- .^-_ LONGITUDK FROM 

*^*^"" aRSSNWICH *'*-AU» GMimrWTCH 

H. M. & H. M. a. 

Albany +4 54 59.99 Denfer +6 59 47.63 

Algiers -0 12 08.55 Edinburgh +0 12 44.2 

Allegheny, Penn .. + 5 20 02.93 Glasgow +0 17 10.55 

Ann Arbor, Mich. + 5 34 55.19 Madison, Wis +5 67 37.93 

Berkeley, Calif. . . + 8 09 02.72 Madrid + 14 45.12 

Berlin -0 53 34.85 Mexico +6 36 26.73 

Bombay -4 51 15.74 New York +4 55 53.64 

Cambridge (Eng.) - 00 22.75 Paris - 09 20.97 

Cambridge (Mass.) + 4 44 31.05 Philadelphia +5 00 38.51 

Cape of Good Hope -1 13 54.76 St. Petersburg ...- 2 01 13.46 

Chicago +5 50 26.84 Washington +5 08 15.78 

16. Give from the table the difference of local times of Green- 
wich, England, and of each of the following places and state 
whether the time is earlier or later than Greenwich time : Albany ; 
Algiers; Ann Arbor; Berlin; Bombay; Chicago; New York; 
Paris; St. Petersburg. 

17. What is the longitude in degrees (°), minutes and sec- 
onds (") of arc of Albany? of Algiers? of Cambridge (Mass.)? 
of Washington? 

18. Which of each of these pairs of places has the earlier local 
time and how much earlier is this time : 

Albany and Allegheny? Chicago and Paris? 

Ann Arbor and Berkeley? Edinburgh and Madison? 

Ann Arbor and Bombay? Cape of Good Hope and Bombay? 

Chicago and Cambridge,Mass.? Paris and Bombay? 

19. Give the differences of longitude in each case of problem 
18. 

20. Solve other similar problems on the table. 



LONGITUDE AND TIME 



303 



§183. Standard Time. 

For convenience of railway traflSc a uniform system of time- 
keeping, known as Standard Time, was agreed upon in 1883 by 
the principal railroad companies of North America. It was 
decided that places within a belt of 15° extending (roughly) 
7^° on each side of the 75th meridian west of Greenwich should 
use the time of the 75th meridian. All places in similar belts 
extending about 7^° on each side of the 90th, of the 105th, and 
of the 120th meridian should take the times of those meridians 
respectively, and hence should have times just 1 hr., 2 hr., and 
3 hr. earlier (less) than the time of the 75th meridian time belt. 
This system has now been generally adopted by most civilized coun- 
tries. In practice, however, the dividing lines of the time belts 
are uregular lines running through railroad terminals. The fol- 
lowing map shows the time belts and the names used to dis- 
tinguish the times of the several belts which cover continental 
United States. In this system the times at all places in the 
United States differ only by whole hours. 



Pacific Time 

120° 



MouifTAiN Time 

105° 



CsNTBAii Time 



Eastern Time 

75° 




FlOUBB 192 



1. When it is 8 o'clock a.m. (Standard Time) at Chicago, what 
is the time at each of the following places : New York? Pittsburg? 



304 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Sb. Louis? Omaha? Denver? Spokane? San Francisco? (See 
the map, Fig. 192.) 

. 2. When it is 9 hr. 10 min. 45 sec. p.m. at Omaha, what is 
the time at New Orleans? Philadelphia? Buffalo? Washington? 
Austin (Tex.)? Boise City? Los Angeles? El Paso? Salt Lake City? 
(Refer to your Geography.) 

3. Answer the questions of problem 2 for 2 : 15 p.m. at St. Paul ; 
for 3: 25 a.m. at Dodge City, Kan. 

4. Answer for other places marked on the map such questions 
as are asked in problems 2 and 3 for Omaha, for St. Paul, and for 
Dodge City. 

The circle SqWr (Fig. 193) represents the earth and the curved 
lines represent the hourly meridians running from the equator, 

qr^ and converging toward the 
poles S and iV. If represents 
some place in the northern hemi- 
sphere, SqJVr is the meridian of 
the place and BQPR may be im- 
agined to represent the meridian 
of the sky {hour circle of the sun) , 
on which the sun, T, is situated. 
The hour circle of the sun in the 
apparatus is held in place at ^. If 
now the crank F is turned so as 
to carry q forward and downward 
FIGURE 193 through E to r, the sun's meridian 

standing stationary, the motions which cause the differences of 
time and the changes of day and night may be understood, by 
recalling that only the half of the globe which is turned to the 
sun is light (in day). /S 6^ iV denotes the prime^meridimi. 

5. As the globe is standing in Fig. 193, what time is it at O? 
at places on the 105th meridian? on the 60th meridian? on the 
15th meridian east* of Greenwich? on the 45th east? 

6. If the meridian SGN is continued around on the other 
side of the globe, what will its number be? 

7. Remembering that any place on the earth, as O, turns com- 
pletely round through 360° in 24 hr., how long does it require 




* Remember that east is the direction toward which the globe is turning, that is east- 
ward means from E toward through r and around to E again. 



LONGITUDE AKD TIME 305 

the Space between any two adjacent meridians shown in Fig. 193 
to pass under the sun's hour circle? 

8. If a man should start from some place on the prime 
meridian (say London) on Friday noon and move westward just 
as fast as the globe turns eastward, what time (by the sun) would 
it be to him during his journey all the way round the globe? 
What hour and day would a Londoner call it when the traveler 
returned 24 hr. later? 

The problem raises the question, ''Where should the traveler 
have changed his date so that his date might agree with that of 
his starting place when he returns?" The answer is, "It has 
been agreed that the date should change at the 180th meridian. " 
When vessels cross this meridian from the east toward the west 
they add a day to their reckoning. If they cross at noon on 
Friday, Friday noon instantly becomes Saturday noon. . Crossing 
from the west toward the east, they repeat a day. In the case 
mentioned, Friday would ''be done over again." The 180th 
meridian is for this reason called the Date Line. Trace it 
round the earth in your Geography. 

9. The date begins on the 180th meridian of longitude and 
travels westward around the earth. Supposing the day begins at 
midnight, give the date and hour on the 180th meridian, when it is 
Wednesday, Oct. 15, 10 P. M., standard time, in Chicago; in 
Washington; in Denver; in San Francisco. 

10. The U. S. Supreme Court has decided that legal time at any 
place is the local time of the place. A man in Akron, 0., insured 
his house at 11:30 A. M. "standard" (90th meridian) time and 
the house took fire at that moment and was destroyed. The 
policy stated that the insurance would be in force at and after 
12 o'clock (noon) of the day of insuring. The difference between 
Akron local time and 90th meridian time is 33 minutes. Would 
the law require the insurance company to pay thv^ policy? Give 
reason for your answer. 

11. If every place on the earth has the time of that meridian 
of the map, p. 296, nearest to it, what will be the time of the fol- 
lowing places when it is 3:25 P. M. in England? (1) Washington? 
(2) Boston? (3) New Orleans? (4) Denver? (5) Rio Janeiro? ((3) 
Cuba? (7) Japan? (8) Philippine Islands? 



306 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



MENSTTBATION 
§184. Areas, Roofing and Brickwork. 

1. Give the rule for computing the area of a square from its 
measured sides; a rectangle; a parallelogram; a triangle. (See 
pp. 102-104 and 179.) 

Fig. 194 represents a plot of the streets and blocks of a part of 
a certain city. The streets are all 75 ft. wide between sidewalks. 
The numbers written on the lines indicate their lengths in feet. 
The dotted lines show how to divide up the areas into parts for 
computation. 




Figure 194 

2. If $240 per foot of frontage on both 3rd and 4th streets 
was paid for block B (Fig. 194), how much did the block cost per 
square foot? per square yard? 

3. Find the area in square yards of other parallelograms, such 
as 0, P, etc., of Fig. 194. 

4. Find the area in square feet of the following triangles of 
Fig. 194: E; G; I; J; K; L; abc. 

Other areas of Fig. 194 require a knowledge of trapezoids. 

1 Definitions. — A four-sided figure (a quadrilateral) like 

Fig. 195. having one pair of parallel sides, is called a trapezoid. 
The altitude of a trapezoid is the distance square across 
Figure i95 between the two parallel sides (called the bases.) 



MENSURATION 



307 



5. Study tlie trapezoids of Fig. 196, and find how to get the 
length of EF^ the line connecting the middle points of the two 
non-parallel sides, from the lengths of the two bases. After com- 
puting EF^ find for each trapezoid a rectangle whose area equals 
the area of the trapezoid. Find the areas of the trapezoids of 
Fig. 196. 



(B^^^t^/K 



Figure 196 

Definition. — The lengths a, 6, and c of the last trapezoid are called 
its dimensions. 

Note. — J the sum of x and y is written \{x-\-y), n times the half sum 
is written \n{x + y). 

6. How do the altitude and the sum of the bases of a trapezoid 
compare with the altitude and base of a parallelogram having an 

^•7 V ^"""'7 V V — ^ area equal to the area of the 

5^-- \ 5^:: ^ ^--^^ trapezoid (Fig. 197)? 

. Figure 197 7. Supposing a, ^, and c 

are the dimensions, in feet, of the trapezoids of Fig. 197, what 
are the areas of these trapezoids? 

8. Find the areas in square feet of C\ of 7>; of J/; of N\ of H 
(Fig. 194). 

9. Calling Z the area of any trapezoid, whose bases are h and 
c and whose altitude is a, write an equation showing how to find Z 
from «, by and c, 

10. Find the area in square yards of ^, ^, N, S, T, Fig. 194. 

Definition. — A square of roofing means a 10' square of roof surface, or 
100 sq. ft. A shingle is said to be laid 4", 4 J", or S'' to the weather 
when the lower end of each course of shingles on the roof extends 4", 4^", 
or b" below the course next above it. 

11. Draw two perpendicular center 
lines, and with the aid of the dimen- 
sions given in the left part of Fig. 198, 
complete an enlarged drawing to a con- 
venient scale, of the development of 
the roof (shown on the right), figurbiss 




308 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



12. If 1000 shingles laid 4" to the weather cover a square of 
roofing, how many shingles will be needed to cover a square, if 
laid 5" to the weather? ^" to the weather? 3" to the weather? 

3r? 

13. The dimensions on the development (Fig. 198) being in 
feet, find the cost of the shingles, at $1.10 a bunch of 250, needed 
to cover the four sides of the deck roof (Fig. 198). Find the 
cost of enough tin to cover the 12' x 16' flat deck at 15^ a 20"x28" 
sheet, the long sides of the sheets being laid parallel to the long 
side of the deck, and allowing 10% loss for joints and overlap at 
edges. 

14. When shingles are laid 4" to the weather, 1000 shingles 
are estimated to cover a square (of roofing). Find the number of 
shingles, laid 4'' to the weather, needed to cover a roof which, 
with gables, is made up of the following parts: 



Shapes Altitudes Bases 

2 rectangles 10' 40' 

2 psvraUelograms 12 ' 26 i ' 

2 parallelograms 12' 14' 

2 triangles 7' 14' 



Shapes Altitudes Bases 
4 triangles 8i' 17' 

2 trapezoids 12 12' and 20)' 
2 trapezoids 12' 5' and 13i' 
8 trapezoids 12' 6* and 14i' 



90c a bunch 



15. Find the cost of the shingles of Prob. 14 
of 250. 

16. Ten years after building this house a new roof was 
put on it. It cost $1.25 per M. to remove the old shingles, 
$3 per M. to dip the new ones, and $2.76 per M. for the labor of 
putting on the shingles. How much did it cost to remove tlie 
old shingles and to re-roof with dipped shingles? 

17. If it takes 14 bricks per square foot of outside surface to 
lay a 2-brick wall, how many bricks will be 
needed to lay the side wall of the build- 
ing shown in Fig. 199, deducting for 5 
windows each 3^' x 6J' and for 5 windows 
each 3 J' x OJ'. 

18. Make other similar problems from 
your own measurements or from dimensions 
obtained from an architect or builder. figure 199 




MENSURATION 



309 



19. The walls of a square brick house extend 6' into the 

ground and rise 24' above the ground to the eaves. From the 

bottom to 12' above the ground the walls are 3 bricks thick and 

from this line to the eaves they are 2 bricks thick. Find the cost 

of laying the bricks of these walls @ $4.25 per M. 

Note. — No allowance for openings of any kind is made in figuring 
the cost of laying bricks in such walls, and outside dimensions are used, 
thus counting corners twice. 

20. From the number of bricks found in problem 21 deduct 

for the following openings : 

Downstairs Upstairs 

1 door 3' 6" X 6' 9" ; 2 doors 8' 6" X 6' 6" : 

1 door 3' X 6' 6" ; 1 window 5' X 6' 6" ; 

2 windows 5' X 6' 6" ; 12 windows 3' X 6' ; 
9 windows 3' X 6' ; 

and 1400 bricks for double counting of comers, and find the cost 
of the bricks @ $12^ per M. 

21. My friend's house has two octagonal (8-sided) towers, the 
roofs being made up of equal triangles (Fig. 200). Find the 
number of slates for both 
roofs, 240 to the square 
(100 sq. ft.), and cost @ 
$2.75 a hundred. 

22. Draw the develop- 
ment of the roof and tower 
to a scale J" to 1', as shown 
in the figure. 

23. From the eaves to 
the ground the towers are 
octagonal prisms, built of 
brick, all faces being rect- 
angles 6' X 40', and each 
prism has fnur faces exposed to the weather. How many 
square feet of brick surface are in the 8 exposed faces, not allow- 
ing for windows? Find the cost of laying the brick @ $6.25 per M. 

24. Deduct for 16 windows, each 3' 6" x 5' 8", and for 4 win- 
dows V 6" X 3', and find the number of bricks in the two layers, 
and their cost @ $14 per M. 




3BB 
BBS 



BBBB 
BBBB 



Figure 200 



310 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



§186. Land Measure. - 

Review §78, pp. 119 and 120. 

The law requires land to be marked out or surveyed in divisions 
of the form of squares and rectangles. In the western states the 
land has been surveyed in accordance with this law. 

To mark out the largest squares, north and south lines, called 
meridians^ are first run 24 mi. apart and marked with comer- 
stones, or by trees, or other permanent objects. East and west lines, 
called base lines, are then run at right angles to these meridians 
at distances 24 mi. apart. This would divide the land up into 
24-mi. squares were it not for the convergence of the meridians 
toward the poles of the earth. Notice this on a map in your 
Geography and on the map of Fig. 192. 

Each 24-mi. tract is then divided into 16 nearly equal squares, 

called townships, by running north and south, and east and west 

lines through the quarter points of the sides of the large tract. 

How long is a township? how wide? how many square miles 

does it contain? 

Certain meridians, called principal meridians^ are run with 
great care, and these principal meridians govern the surveys of 
lands lying along them for considerable distances both toward 
the east and toward the west. The tiers, or rows, of townships 

running north and south along 
the principal meridians are 
called ranges. The first tier on 
the east is called range No. 1 
east, and is written KIE; the 
second range is No. 2 east, 
written E2E, and so on. 

Point out on the drawing. 
Fig. 201, EIW; E2W; R3W; 
E2E; E4W. 

Certain base lines are run 
with great care and are called 
standard base lines. The rows 
of townships running east and 
west are numbered with refer- 
ence to these standard base 
lines. A township in the first row north of a base line is town- 
ship No. 1 north, and is written TIN; one in the second row 
south is called township No. 2 south, written T2S, and so on. 

Interpret the following symbols and point out on the drawing, 
Fig. 201, the townships indicated: T3N; T4N; TlS; T2N. 





a: 






Standi 


Ul Ul 

rd Line 






4 


4 


4 


4 


4 


4 










c 








c 


3 


3 


3 


•1 


3 


3 






2 


2 


2 




2 


2 






1 


1 








1 


-- 






Ba 


se Lin 


e 




f 


I 


1 


Fig 


UBE 9 


1 


1 


J 



MENSURATION 



311 



A township is identified by giving its number and range from 
some standard base and principal meridian. 

Point out these townships on the drawing, Fig. 201: TIN, 
R2W; T3N, RlW; T4N, R2E; TlS, R3W. 

The law also requires 
townships to be sub- 
divided into smaller 
squares, called sections. 
Sections are numbered, 
beginning at the north- 
east corner and run- 
ning toward the west 
to 6, then 7 is just 
south of 6, and so on, 
as in Fig. 202. 

Sections are then sub- 
divided into quarters (see 
section 16), half-quarters 
(see sections 13 and 26), 
and quarter-quarters (see 
section 29). 

1. Referring to Fig. 
202, read and write the 
descriptions of the divi- 
sions of section 16; of section 26; of 13; of 29. 

Whatever deviations there may be from exactly 640 A., in the 
sections of any township, due to convergence of meridians or other 

causes, are required by law 
to be added to, or sub- 
tracted from, the north and 
west rows of half-sections. 
These tracts are then not 
called fractional sections, 
but are called Zo^5, and are 
numbered in regular order 
as shown in Fig. 202. 

A section then always 
means exactly 640 A. Any 
fractional part of a section 
means the ' corresponding 
fractional part of 640 A. 
The lot, on the contrary, 
must always be measured 
FiGuaa 203 before its area is known. 



Lot 


16 


Lot 15 
321 A 


Lot I4t 


iLot 13 


Lot 12 


^U' 


^% 


6 


5 


4 


3 


2 


1 




7 


8 


9 


10 


II 


12 


S5 

>s 


18 


17 






15 


14 










h 




I 


p 


i 


19 


20 


21 


22 


23 


24 




30 










28 


27 






25 




-I 


n 




^ 


ft 




t 


V 




^ 


















3i 


32 


33 


34 

1 


35 


36 



FIGURB 202 



L48 
38 

J.CH 



L47 
41 

URCH 



L49 

4-IA 

J. 



L50 
82 A 



40 80 

Q^g.M. EVANS 



LSI 
8IA 

*«|S.PBRRY 



8iA 
H.BE 



JbWQSh 



L 46 

79 A 

W. BROWN 



80 



80m 160 



Seel ion Line ^ 



^^l^ 



160 



80 
VENS 



C.PAfKS 
160 



D Line 



L45 

79 A 

J. BRADEN 



60 g40 
H. BRADEW . 



L44 
39A 



CI 



160 



t 



160 



E80 



isO 



160 



80 

> 

580 



L43 
38A 



80 



o 
-C80 



312 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

2. Following is an assessment list of farm property. Fill the 
blanks from Fig. 203: 

OWNBB DESCBIPTION NO. ACBBS 

H. Peabody EJ SEJ Sec. 8 

H. Peabody E J NEJ Sec. 8 

J. White EJ SEJ Sec. 5 

J. James SE^ NE^ Sec. 5 

J. James Lot 43 .... 

0. Gibson '. Lot 44 .... 

0. Gibson SWi NE^ Sec. 5 .... 

3. Make out an assessment list for all owners in Sec. 6. 

4. Make and solve a similar problem for owners in Sec. 7. 

5. Correct the mistakes in this erroneous assessment list: 

OWNKB DESCBIPTION NO. ACBI8 

S. Perry WJ NWJ Sec. 7 81 

J. Hay Wi SWi Sec. 6 88 

J. Hay Wi SWi Sec. 7 80 

H. Ochiltree SWJ NW^ Sec. 6 41 

H. Ochiltree SEJ^ NE J Sec. 6 40 

H. Ochiltree EJ SE} Sec. 6 80 

J. Church NWi NWi Sec. 6 38 

J. Church NEJ NWJ Sec. 6 41 

§186. Volumes. 

Definition. — The volume of any figure is the number of cubical units 
within its bounding surfaces. 

1. Give the rule for finding the volume of a square prism 
(called also a rectangular parallelepiped) . (See pp. 112, 113.) 

2. Give a rule for finding the volume of an oblique parallele- 
piped (Fig. 148, p. 277) having the same base and the same 
altitude as a given rectangular parallelepiped. (See Figs. 150 
and 151, pp. 277 and 278.) 

3. How would you find the volume of a hollow beam 12 ft. 
long, halving a cross section like (1) Fig. 05, p. 132? (2)? (3)? 
(5)? (6)? (4)? (9)? 



MENBURATION 



313 



Scale 1 :6 



4. Giv-e the volu-mes of the beams of problem 3, if the length 
of the beam in each case is I feet. 

5. A model of a square prism (like Fig. 145) 
having a base 2 in. square and an altitude of 8 in. 
will contain how many cubic inches of sand? 

6. The model of a right circular cylinder, 
made as shown in the scale drawing of Fig. 204 
and pasted along the flap, DF, and around one 
end with a strip of paper, was filled with sand 
and poured into the empty model of the square 
prism of problem 5. It filled the square prism 
a little more than f full. About how many cubic figubb ao4 
inches are there in the model of the cylinder? 

7. It is shown in geometry that the volume of any right cir- 
cular cylinder is. 7854 (= j) times the volume of a square prism 
of the same altitude and having for one side of its base the 
diameter of the cylinder. Find the volumes of these circular 
cylinders : / 



£ 



(1) Diameter 4", altitude 6" 


(4) Diameter ISf, altitude 8" 


(2) " 7" " 7" 


(o) " 6' " 10.5' 


(3) " 10" '♦ Sf 


(6) " 2r " a 



8.. How long is AB? How many square inches in the rectangle 
ADFG (Fig. 204)? How many square inches are there in the en- 
tire outside surface of the cylinder? 

9. The inside diameter of the cylindrical water tank of a 
street sprinkler is 3.6 ft., and its length is 10 ft. ; how many liquid 
gallons does the tank hold? 

10. The tank of the sprinkler is filled through a hose of 2^' 
inside diameter from a hydrant from which the water flows at the 
rate of 300 linear feet per minute. How long will it take to fill 
the tank? 



Suggestion. — In 1 min. a cylinder of water as large as the inside of 
the hose and 300' long flows into the tank. 

11. Find the weight of an iron rod ^" in diameter and 24' long, 
if iron weighs 450 lb. per cubic foot? 



314 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

12. How many cubic inches of air are there in the hollow tire 
of a bicycle wheel 28" in diameter (from center line to center line 
of tubes), the hollow having a diameter of If"? 

13. How many cubic inches of rubber are there in the hollow 
tire of a 32" automobile wheel if the inside diameter of the cylin- 
drical tube is 3" and the outside diameter is 4 inches? 

^ ^ 14. Fig. 205 is a scale drawing of a pattern 

"IT for the paper model of a cone whose base is to 

^^ be a circle 2" in diameter and whose sloping 

^ side from the apex to the base is to be 8". The 

\ arc AHB is just as long as the circumference 

\ of the base. Compute the length of the cir- 

^^^^^^^3^ cumference of a circle whose radius is 2", and 

Scale i^/Si^ ^^ another whose radius is 8", and find the 

vJ/ ratio of the first to the second. 

Figure 205 15^ What part of the whole circumference 

(with center C) is the arc AHBl What part of 360'' is the angle 

AGBl 

16. Make a right angle, as HOX^ and bisect it (See Problem II, 
p. 176). CD is the bisector. Bisect angle HCD and obtain CB. 

17. With C as center and with a convenient radius, as CE^ 
draw the indefinite arc PNME. Put the pin-foot on G and mark 
an arc across arc OP at N. Draw CiV and prolong it. This makes 
angle GCN equal to angle GCM. Now with (7 as a center and 
with 8" as a radius, draw the arc AHB. Prolong CHy make 
HO = 1", and draw the lower circle. Provide the flap and paste 
up the model of the cone. 

If such a model is carefully made and the height is measured 
and if the model of a cylinder has the same height and the same 
base, the model of the cone will be found to hold ^ as much sand 
as does the model of the cylinder. It is proved in geometry that 
the volume of any cone equals ^ of the volume of a cylinder having 
an equal base and an equal altitude. 

18. Find the volumes of these circular cones: 

(1) Diameter of base 3", altitude, T' 

(2) " " 6" " 20" 

(3) " '' 6.8" " 2.25' 

(4) '' '' 3.95" " 10.25" 




MENSURATION 315 

19. How many cubic inches of water are there in a funnel- 
shaped vdssel, if the water is 4" deep and the diameter of the 
surface of the water is 4.5"? (See Fig. 206). 

20. How many cubic inches of water will it 
take to fill the same vessel to twice the depth, or 
8"? (See Fig. 206.) 

21. How many cubic inches had to be poured 
into the vessel to fill it to 8" depth if the depth of 
the water was 4" at the beginning? figubk aoe 

22. A mountain peak has the shape of a circular cone whose 
altitude is 1.25 mi. and the diameter of whose base is 2.35 miles. 
Find its volume in cubic miles. 

23. How much water will it take to fill a conical vase 10" 
deep and 3.225" across at the top. 

24. The conical tower of a building is 24.75' across at the 
base and 36.8' high. How many cubic feet of space does it 
occupy? 

25. If a cord or waxed tape (bicycle repair tape) be wrapped 
around the curved outside surface of a half croquet ball, as a top is 
wound, until the surface is covered, and then if the same cord, or 
tape, be wrapped around on the flat circular base beginning at the 
center until the circle is covered, the length of the former cord 
will be found to be just twice the latter. The area of the surface 
of the wliole ball is then how many times the area of the circular 
section of the ball? What radius has this circular section? 

It is proved in geometry that the area of the surface of a sphere 
equals 4 times the area of a circular section going through the 
center of the sphere. 

26. The radius of a ball is 1^"; what is the area of the 
greatest circular section of the ball? What is the area of the 
surface of the ball? 

27. Measure the circumference of a baseball and find how 
many square inches of leather there are in the cover of the ball. 

28. How many square inches of paint will it take to cover the 
surface of a globe of 10" radius? of 10" diameter? 

29. The average diameter of the earth is 7918 mi. ; how many 
square miles are there in its surface? 




316 BATIOlSrAL GRAMMAR SCHOOL ARITHMETIC 

30. Calling s the surface and r the radius of a sphere, write 
an equation showing the relation between s and r. 

It is seen on p. 277 that the volume of a triangular right 
prism equals i the volume of a square prism whose base and alti- 
tude are equal to the base and the altitude of the square prism. 

31. Find the volume of a right 
triangular prism whose altitude is 18" 
and whose base is a triangle having 
a base of 8" and an altitude of 5" 
inches. 

FIGUBE207 PIOUBE808 j, ^ triaugular pyramid be care- 
fully modeled (Fig. 207) and filled with sand it will be found that 
just 3 times the volume of the pyramid is equal to the volume of 
the model of a triangular prism (see Fig. 208) of equal base and 
equal altitude. It is proved in geometry that the volume of any 
pyramid equals i of the volume of a prism having an equal base 
and an equal altitude. Notice in Fig. 208 how a triangular prism 
may be completed on a triangular pyramid having the same base 
and the same altitude as the prism. 

32. Calling V the volume, B the area of the base, and a the 
altitude of a triangular pyramid, write an equation, showing the 
way V would be computed from B and a. 

33. Find the volume of a pyramid whose base contains 16 sq. in. 
and whose altitude is 12 inches. 

34. The Great Pyramid of Egypt is 481 ft. high and its base is 
a 756' square. If it were solid and had smooth faces, how many 
cubic feet of masonry would it contain? 

35. At the close of the nineteenth century the United States 
had 195,887 mi. of railroad. How many times would these rail- 
roads, if placed end to end, encircle the earth? (Use v = 34^, and 
the radius of the earth = 3959 miles). 

36. The ties used for these roads would contain wood enough 
to make a pyramid of the same shape 1395' high with a 2192' 
square for its base. How many cubic feet of wood were used for 
the railroad ties? 

37. It has been computed that the materials used for the road 
beds for these railroads would make a solid pyramid 2470 ft. high 



COKSTRUCTIVE GEOMETRY 



317 



How many cubic feet 




and having a 3870 ft. square for its base, 
would this make? 

If the entire surface of a globe, or sphere, were divided up 
into small triangles like the one shown in Fig. 209, and the sphere 
were cut up by planes cutting along the curved sides 
of the triangles and passing through the center, 0, 
the volume of the sphere would be divided up approx- 
imately into small triangular pyramids, having their 
vertices at the center. The volume of each pyramid 
would be the product of its triangular base by J of the figure 209 
the radius of the sphere. The sum of the areas of all 
triangular pyramids would equal the surface of the whole sphere 
and the sum of all the volumes of the pyramids would equal the sur- 
face of tlie whole sphere multiplied by i of the radius of the sphere. 

38. Calling V the volume and r the radius of a sphere, give 
the meaning of the formula : 

o o 

39. Find the number of cubic inches in a sphere of 2" radius. 

40. How many cubic inches in a croquet ball 4" in diameter? 
in a tennis ball 1.75" in diameter? in a baseball 2.1" in diameter? 
in a globe 10.15" in diameter? 

41. Calling t^e sun, the moon and the planets all spheres with 
diameters in miles as rn the following table, compute their cir- 
cumferences in miles, their surfaces in square miles and their 
volumes in cubic miles : 



Moon 2,160; 

Mercury 3,030; 

Venus 7,700; 

Earth 7,918; 



Mars 4,230 

Jupiter 86,500 

Saturn 73,000 



Uranus 31.900; 

Neptune 34,800; 

Sun 866,400, 



§187. Constructive Geometry. 

Problem I. — To find the center of a. given arc. 

Explanation. — Let AB^ Fig. 210, be the given arc whose center is to 
be found. 

Mark any three points, as C, E, and D, on the 
arc. Draw the straight lines CE and ED. Bisect 
each of these lines as in Problem VI, pp. 100 and 101, 
and prolong these bisectors until they intersect as 
at O. O is the required center. 

Definition. — The lines C-EJand ED, each of which 
connects two points of the arc, are called chords of. 
FiGUBB 21U the arc. 




318 



EATIOXAL GBAMMAR SCHOOL ARITHMETIC 



To solve this problem is it necessary actually to draw the 
chords? 

How could you find the center of a circle that would go through 
any three points not in a straight line? Mark 3 points not all in 
the same straight line and draw a circle through them. 

Problem II. — To bisect a given arc. 

Explanation.— Let AB, Fig. 211, be the given arc. 

With ^ as a center and then with B as a center and 
with a radius greater than the distance from A to the 
middle of the arc AB^ draw the dotted arcs as indicated. 
Lay a ruler on the intersections of the two arcs and draw 
a short line across the arc, as at C. C is the mid-point of 
the arc, and arc -4C = arc CB. 



X 



v 

FIGUBB 211 



Problem III.- 
triangle. 



-To circumscribe a circle around an equilateral 



Explanation.— Draw an equilateral triangle 
as in Problem VII, p. 102, and bisect two of its 
angles as shown in Fig. 212. With the intersec- 
tion of the bisectors as a center and with a radius 
equal to the distance from this intersection to 
any vertex, draw a circle. This circle is said to 
be circumscribed around the triangle. 

Problem IV. — To draw a trefoil. 




Figure 212 




Figure 213 



Explanation.— Draw an equilateral tri- 
angle and bisect one of its sides, as shown in 
Fig. 213. With the upper vertex as center and 
with a radius equal to the distance from this 
vertex to the middle of the bisected side draw 
an arc of a circle around until it touches the 
sides of the triangle both ways. Draw arcs 
around the other vertices in the same way. 

If desired, other circles, with slightly 
longer radii, may be drawn just outside of 
these until the arcs come together but do not 
cross. 



Problem V. — To construct a square and to 
draw a quatrefoil (a four-foil) upon it. 

Explanation.— Prolong BA through A far enough 
to draw a perpendicular, as AC, to AB at A. Draw 
this perpendicular and make A(J =^ AB With AB as a 
radius and (1) with C as a center, then (2) with ^ as a 
center, draw two intersecting arcs at D. Draw CD 
and BD and complete the drawing as shown in Fig. 214. 




Figure 2U 



CONSTRUCTIVE GEOMETRY 



319 



Problem VI. — To draw the designs of Fig. 215. 



Explanation. — Draw a square like the 
dotted squares'of Fig. 215. Study the two draw- 
ings, decide where the centers of the arcs are 
and complete the designs. 




Figure 215 



Problem VII. — To draw a sixfoil. 

Explanation.— Draw a circle, like the dotted 
one in Fig. 216, with a radius as long as one side 
of the regular hexagon is to be. Draw the hexa- 
gon (see Problem XI, p. 104). Bisect a side of the- 
hexagon, and, using the vertices of the hexagon 
as centers, complete the sixfoil, as shown. 

Outside arcs may be added if desired. 




FlGUBB 216 



Problem VIII. — To construct a right triangle. 



The symbol 
"perpendicular 
ular to." 



1 means 
to," or 



'perpendicular," 
"is perpendic- 



CoNSTRUCTioN. — Draw the line CDlAB, 
Fig. 217. Then draw the line OF connecting 
any point, as G, of the line CD with any 
point of AB, as F. 

Definitions. — The longest side, that is, 
tlie side opposite the right angle of a right 
triangle, is called the hypothenuse. 




Figure 217 



What side of the triangle, GOF, is the hypothenuse? What 
angle is the right angle? 




FXOUBB 218 



Problem IX. — To construct a right 
triangle having a given hypothenuse. 

Construction. — Let the line AB, Fig. 218. 
5 denote the given hypothenuse. Bisect AB 
as at O and with O as a center and OA as 
a radius, draw a semicircle. Connect any 
point of the semi-circumference, as E, D, or C, 
with A and with B. Any such triangle is a 
right triangle and the line AB is its 
hypothenuse. 



320 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



218. 



Point out the right angle of each of the three triangles of Fig. 




Figure 219 



Problem X. — To construct a right 
triangle, having given the two sides which 
include the right angle. 

A « Construction.— Let the two given sides 

'^—^ be a and b, Fig. 219. 

Draw BC 1 DE, and make OA==a and OB 
= b. Connect A with B, BOA is the re- 
quired triangle. 

How may an isosceles right triangle 
be constructed? 



Problem XI.— To find the relation of the squares of the 
sides of an isosceles right triangle. 

Construction.— Construct an isosceles right trian 
gle and on each of its three sides draw a square. Draw 
the dotted lines and cut the side squares as shown in 
Fig. 220. Fit the pieces over the large square on the 
hypothenuse. If the area of each side square were 9 
sq. in. what would be the area of the large square? 

FIGURB 220 

Problem XII. —To find the relation of the squares of the 
three sides of any right triangle. 





F16URB 221 

Construction. — Construct any right triangle and then construct a 
square on each of its three sides. Cut the side squares as shown.* Fit 
the pieces over the s^quare drawn on the hypothenuse, as indicated by 
the dotted lines in Fig. 221. What single square has an area that equals 
the sum of the areas of the squares on the two shorter sides of any right 
triangle? 

^he center of a square Is the lutersectiou of Us diagonals. 



SQUARES AND SQUARE ROOTS 321 

1. Denoting the length of either short side of Fig. 220 by a, 
what denotes the area of the square drawn upon this side? 

2. Denoting the length of the hypothenuse of Fig 220 by /*, 
what denotes the area of the square drawn on the hypothenuse? 

3. Write an equation from Fig. 220, showing the relation 
between a* and A*. 

4. Denote the lengths of the three sides of any right triangle 
(Fig. 221) by a, 5, and h (A being the hypothenuse). What will 
denote the areas of each of the squares on the three sides? 

5. Write an equation showing the relation of the squares of 
the sides of any right triangle. 

PROBLEMS 

1. The sides of a right triangle are 3" and 4"; what is the 
length of the hypothenuse? 

SUOOESTION.— 7i2 = 3* + 42 = 9 -f 16 = 25 ; what is the value of h? 

2. The hypothenuse of a right triangle is 10", and one of the 
sides is 6" ; what is the other side? 

SUOGESTION.— lO^ = a2 + 6^ ; find the value of a? 

3. The sides of a right triangle are denoted by a, J, and the 
hypothenuse by h; find the unknown side in each of the following 
right triangles: 

(1) a = 9, 5 = 12; (4) a = 32, 5 = 24; 

(2) a = 12, h = 20; (5) b = 21, A = 35; 

(3) b = 15, h = 25; (C) b = 27, h = 45. 

Before the hypothenuse of a right triangle, whose sides are 34 
and 26, can be computed, it is necessary to know how to find the 
square roots of given numbers. 

§188. Squares and Square Boots. 

DKFiNmoN. — The product obtained by using any number twice as a 
factor is called the sqvure of that number Thus, 36 is the square of 6, 
because 6, used twice as a factor gives 36 (6x6 = 36). The square of a 
number as 6 is often written thus, 6^. What does the small 2 show? 



322 RATIONAL GRAMMAE SCHOOL ARITHMETIC 

The following squares should be committed to memory : 



Squares of Units 


Squares of Tens Squares of Hundreds 


1^= 1; 


10* = 100; 100*= 10000; 


2«- 4; 


20*= 400; 200*= 40000; 


3«= 9; 


30* = 900; 300*= 90000; 


4* = IG; 


40» « 1600; 400* = 160000; 


5««25; 


50« = 2500; 500* = 250000; 


6» = 36; 


60« = 3600; 600* = 360000; 


7« = 49; 


70« = 4900; 700« = 400000 ; 


8« = 64; 


80« = 6400; 800* = 640000; 


9« = 81; 


90* = 8100; 900* = 810000. 


1. Write all the 


5 pairs of number^ which, multiplied together, 


give the product 


36; the product 16; the product 64; the 


product 49. 




Note.— Write the 


pairs of factors of 86 thus : 1 and 36 ; 2 and 18 ; 3 and 


12; 4 and 9; 6 and 6. 


Proceed similarly with the rest. 



Definition.— The square root of a number is one of the tico equal 
factors of it. The sign of square root is -s/, called the radical sign. Thus, 
y/'Z^ means the square root of 25, which is 5. 

2. Give the square roots of the following numbers: 

9; 16; 49; 64; 81 ; 400; 2500; 3600; 160000; 490000; 810000. 

To find the square root of a number not in the table above, it 
is necessary first to learn how the square of a number is formed 
from the number. 

From the table answer the following questions: 

3. How many digits are there in the square of any number of 
units? of tens? of hundreds? 

4. Find the square of each of the following decimals: .1, .3, .5, 
.7, .9, .01, .03, .04, .06, .07, .09, .001, .003, .005, .006, .007, .009. 

5. How many decimal places are there in the square of any 
number of tenths? of hundredths? of thousandths? 

6. If, then, the square of any number of units contains only 
units and tens, what places of any number that is a square must 
contain the square of the units of its square root? 

7. What places must contain the square of the tens? of the 
hundreds? of the tenths? of the hundredths? of the thousandths? 



SQUARES AND SQUARE ROOTS 



323 



8. If, then, we separate a number, whose square root is desired, 
into two-digit groups, beginning at the decimal point and proceed- 
ing both toward the left and toward the right, what one of 
these groups must contain the square of the units of the square 
root? the square of the tens? of the hnndreds? of the tenths? 
of the hundredths? 

9 Find the square of 46. 

Meaning of Common Method 
46 = 40 + 6; 

462 = (40 + 6)2 =(40+6) (40+6) 
= 40X40 + 40X6+6X40 + 6X6 
= 402 _|_2 X (40 X 6) + 62 = 1600 + 
480+36 = 2116. 

Show the meaning of the parts of 
this sum in Fig. 232. 

Thus it is seen that the square 
of a two-figure number is the sum of (1) the 
square of the tens, (2) twice the product of the 
tens by the units, and (3) the square of the 
units. 

This shows that if any two-figure number be denoted by f + w, where 
t denotes the tens and u the units, the square is formed thus : 

t + u ' ^ 

t + u 




2116 
= 2116. 



FlGUBl^ 222 



40+6 

40+6 

240 + 36 
1600 + 240 

1600+480 + 36 



Show the meaning of the equation 
Fig. 223. 



tu + u^ 
t^ + tu 

by 




Figure 223 



10. Find the square root of 2116. 

Convenient Form 
46 = required root ; 
2116 denoted by t^ + 2tu + u^ ; 
1600 = greatest square (of the table) in 2116; 
2f = 80 516 contains 2tu + u^, where ^ = 40 ; 
2^ + u = 86 516 denoted by 2tu + u^ , wliere m = 6 ; 



Check: 46 X 46 = 2116. 

Shortened Form 
21 '16 I 46 = required root 
80 16 

CJieck: 46x46 = 2116. 



86 



516 
516 



324 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



11 Find the square roots of the following numbers: 

(1)484; (3)1156; (5)1296; (7)7569; (9)9604; 
(2) 729; (4)1225; (G) 5G25; (8) 9409; (10) 110224. 

12. Find the square root of 2079. 3G. 



80 
5 

Hn 

uo. 

^.6 

90.6 



20'79.86' 
16'00 



45.6 



4'79. 
4'25. 



54.36 
5436 



Ans. -^2079.36 = 45.6 
Check: 45.6 X 45.6 = 2079.: 



CONVENIEKT FORM 

Explanation. — Separate the number 
into two-digit groups. Beginning on tbe 
extreme left, subtract the greatest square 
of tens in 2iO hundreds, viz., 16 hundreds 
(=40^), and write 4 tens on the right as 
the first root digit. Double the 4 tens and 
use the result 80 as a trial divisor. 80 is 
contained in the remainder, 479, 5 times. 
Write 5 as the 2d root digit and also add it 
to the 80. giving 85 as the complete divisor. 
Double the part of the root found, 45, 
giving 90 for the next trial divisor and 
complete the steps as before. 



13. Find the square root of 1578 to 3 decimal places. 









Convenient Foem 


15'78.00'00'00M39.724 + 





60 
9 


678. 






66 

78.0 
.7 


621. 
1 57.00 




Explanation.— Annex zeros and 
proceed as before. 


s:7~L 

7»v40 
.02 

79.4a 


55.09 
1.9100 




Check: 39.724 X 39.724=1577.996176. 
rem. = .003824. 


1.5884 




1578. 


79.440 


.321600 






79.444 




.317776 







.003824 remainder. 
In actual practice the decimal point is needed only in the root 

14. Find the square roots of the following numbers: 

(I)- 1900.96; 

(2) 4719.G9; 

(3) 5055.21; 



(4) 61.1524; 

(5) 75.8641; 
(G) 79.9236; 



(7) .5476; 

(8) .458329; 

(9) 1.216609.* 



SQUARES AND SQUARE ROOTS 325 

15. Find to 3 decimal places the square roots of the following 
numbers : 



(1) 5; 


(4) 


.85; 


(7) 1683; 


(10) 1.85; 


(2) 7; 


(5) 


.25; 


(8) 6875; 


(11) 26.79; 


(3) 15; 


(6) 


125; 


(9) 7328; 


(12) 64.893 



16. Find by multiplication the values of the following expres- 
sions : 

(1) (if; (3) (*)'; (5) (^)'; . (7) {±r; 

(2) (f)^ (4) (if)'; (6) (H)'; (8) (^)'. 

17. Make a rule for finding the square root of a common 
fraction. 

18. Find, without reducing the common fractions to decimals, 
the values of the following expressions and prove them by multi- 
plication: 

(1) ^f; (4) ^TI|; (7) v/^v;; 

(2) y/h (^) v/^5 (8) ^/4H> 

(3) v/T|; (6) ^^; (9) p 

19. The sides of a right triangle are respectively 34" and 26" 
long, how long is the hypothenuse? 

20. The center pole of a circus tent is 35' high, and a guy 
rope is stretched from the top of the pole to a stake 56' from the 
bottom. How long is the rope, supposing the ground level and 
the rope straight, allowing 4' for tying? 

21. 50' of the top of a tree standing on level ground is broken 
by the wind and remains fastened to the stump. If the top 
strikes the ground 30' from the stump, how high was the tree? 

22. A horse is staked out by a rope 40' long to the top of a 
stake 15" high. Over what area can the horse graze? 

23. The vertical mast of a hoisting derrick 35' high is held 
in position by four guy ropes staked to the ground at the four 
corner points of a square. The stakes are 75' from the bottom 
of the mast. How much will the rope for the 4 guys cost at 2.5^ 
a foot, 20' being allowed for knots? 

24. What is the area of the square whose corners are at the 
stakes? 



326 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 





25. Find Ihe length of the diagonals of a rec- 
tangle 48' X 64' long. 
/i^ 26. A railroad runs diagonally across a rectan- 

FiouEE 2-:4 gulap- farm 120 rd. x 160 rd. If each rail is as long 
as a diagonal of the farni, how many yards are there in "the rails? 

27. The center pole of a circus tent is held by 6 ropes tied to 
stakes at the corners of a regular hexagon (Fig. 44, p. 104). The 
ground is level, the pole is 48' high and a side of the hexagon 
is 64'. How many feet of rope are needed, if the knots take ISO'? 

28. The sides of an equilateral triangle are 20' 
long. A straight line drawn from any vertex to the 
middle of the opposite side is perpendicular to this 
side and bisects it. What is the area of the triangle? figure l2.i 

29. One side of a regular hexagon is 36'. 
Find the area of the hexagon? 

30. A straight line drawn 
from the vertex of an isosceles 
triangle to the middle of the , 
base is perpendicular to the\ 

FIGURE 226 base. Find the area of a 

regular pentagon whose sides are 24", if the radius 
of the circular drawn around the pentagon is 18". figure 227 

§189. Cubes and Cube Boots. 

1. What is the volume of a cube whose edge is 5"? 6"? 7"? 

11"? 18"? 21"? 24"? 

Definition. — The cube of a number is the product obtained by using 
the number 3 times as a factor. Thus the cube of4is4x4X4 = 4' = 64. 

Table of cubes of numbers to be learned: 





Cubes or Units Cubbs op Tbnc 


P= 1' 


10' = 1000 


2^= 8 


20' = 8000 


3^= 27 


30'= 27000 


43= 64 


40^= 64000 


5^ = 125 


50=* = 125000 


6^ = 216 


; 60' = 216000 


7» = 343, 


70' = 343000 


8» = 512' 


80' = 612000 


9» = 729; 


90' = 729000. 



CUBES AND CUBE R00T8 



327 



Definition. — The cube root of a number is one of^its 3 equal factors. 
Cube root is indicated by the sign y^ , Thus, i/^729 means one of the 
three equal factors of 729, which is 9. The sign ^/^ is called a radical sign. 

2. Between what two whole numbers are the following cube 
roots : 

u^lO? ^/35? ym ^^450? ^/075? ^895? ^^ 58000? ^^ 480000? 

3. Find by multiplication the values of the following cubes: 
(3.5)»; (6.7)»; (13.5)»; (i)'; (|)»; (|)»; (^V)'; (33J)»; (ISf)'. 

4. Make a rule for finding the cube of any common fraction. 

5. Prove the following relations by multiplication: 

^2m = 13; ^4096 = 16; ^M^h ^^ = t\; 
^IIH = H; ^27a»=3a. 
G. Make a rule for finding the cube root of any common 
fraction. 

7. Find the cube roots of the following: 

4; II; m; -eWV. iV«V; TiWinr. 

§190. Triangles Having the Same Shape (Similar Triangles). 

1. Write the numerical values of the follow- . 
ing ratios from Fig. 228: 

^^^AC- ^^^BO^ ^^^AB^ (^>^.' ^^^AO' 
(«)^' (^)T6'' (^)t^= («)j^- 

♦ The length of ab is l^^ = V^^ Why? See Prob. XII, p. 320. Figure 22« 

2. In Fig. 229 the triangles have the same shape. Write 
the values of the ratios : 





■ 








1 


3 










i^ 













^ 








V 


K* 


^ 








u- 




^ 


J ' 






f 


Ih 










^L 










a 




f 











=="- s 


-;fe:: 


S^ T 


?* t ^ ^- 


=:;2^::-: 




^ _; , 



^ ' BO 



P).-?^r 



(3) 






3. In Fig. 230 the triangles 
have the same shape, and AC = 
7 X ac. Find the sides of the 
triangle ABC^ if ac= ^V; ab = 
r, and *c = if'. 



^ B 


^'^ -^ 


^^ 


^^^ 


— ' ,^ 


]^ 


4r J 


1^^ 


L "^!^ 


^4~-^^c- 



Figure 229 



Figure 230 



328 



KATIOKAL QKAMMAR SCHOOL ARITHMETIC 



4. Supposing that ac {^^\ AC) represents 7' (Fig. 230), al, 
10' and Ic^ 11', what lengths do AC^ AB, and BC represent? 

5. In Figs. 231 and 232,AB = ^ab; if ^^ represents 1 mi., ^C, 
3 mi., and -4C, 3J mi., what distances do ab, hc^ and ac represent? 



, 








b 




















B 




j/ 






c 




















^ 


r:r: 


■** 


c 














^ 


/ 




















n 
























V 




FiGuaE 231 



Figure 232 



6. Draw a triangle having sides of 1^', f", and i'\and another 
having sides of 3", 2 J", and IJ". Call the angles opposite the 
sides 1", f", and J", a, J, and c, respectively, and the angles 
opposite the sides 3", 2 J", and IJ", A^ B^ and C, respectively. 
Do these triangles have the same shape? Carefully cut out the 
triangles and place angle a over angle A ; then angle h over angle 
B\ and, last, angle c over (7? What do you find to be true in each 
case? 

Definition.— In triangles having the same shape, angles lying oppo- 
site proportional sides in different triangles are called corresponding 
angles. 

It. Read the pairs of corresponding angles in Figs. 228, 229, 
230, and 232. 

8. In triangles having the same shape, how do corresponding 
angles compare in size (see Problem 6) ? 

9. In Fig. 228 find the ratio of the side lying opposite the angle 
B in triangle ABO to the side lying opposite the corresponding 
angle in triangle abc. Find a similar ratio between any other pair 
of such sides. How do the ratios compare? 

10. Answer similar questions for the triangles of Figs. 229, 
230, and 232. 

Definition.— In triangles having the same shape, sides, as AB and 
ah (Fig. 232) or BC and 6c, that lie opposite the equal angles are called 
corresponding sides. 

11. In triangles having the same shape state a law of relation 
of corresponding sides. 



USES OF SIMILAR TRIANGLES 



329 




FlOURS 233 



12. Triangles ABC and ade (Fig. 233) have th' 
same shape. How do their corresponding anglei 
compare? What relation holds for their cor- 
responding sides, i.e., how do these ratios compare : 
ABiad? BC'.de? ACiae? With a protractor 
measure each pair of corresponding angles. How 
do they compare in size? 

13. What is the ratio of the areas of triangles ABC Skud ade 
(Fig. 233)? Hotv may this ratio he found from the ratio of a pair 
of corresponding sides, as AB and ad? 

14. Draw a pair of triangles, one having sides of 1", 1^", and 
If", and the other having sides of 4", 6", and 7". Give the ratio 
of each pair of corresponding sides; the ratio of the areas of 
the triangles? How may the ratio of the areas be computed from 
the ratio of a pair of corresponding sides? 

15. State a law connecting the ratio of the areas of triangles 
having the same shape with the ratio of any pair of corresponding 
Bides. 



§191. Uses of Similar Triangles. 



HOUSE 



-tB 



In Fig. 235 MNQR represents an 
18"x20" board, provided with a mov- 
ing arm, OE^ and a protractor. 
The arm is made of light wood. A 
_^N slot ES is cut away and a thread is 
stretched on its under side and fastened 
at the ends. The ends of the thread 
are sunk into the wood so as to permit 
the thread to move smootlily over the 
protractor as the arm CO is turned 
round 0. A common pin stuck ver- 
tically at C and at completes the 
apparatus. The pin at should be driven through far enough 
to fasten the arm to the board at as a pivot.. The protractor 
is held by thumb tacks at 7, T. The board may be placed on a 
window sill, or on a post in a fixed position, while measure- 
ments are being made. It is desired to measure the distance from 
A to B^ A and B being on opposite sides of a large building. 

Use the proportion form of statement in all problems, calling 
the length of the unknown line x. Then find the value of x. 




Figure 234 



Figure 235 



330 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Solve some problems like this from your own measures. Build- 
ings, steeples, hills, trees, furnish good problems. 

1. The board was placed on a window sill and the arm set in 
such position, FO^ that when the eye sighted along the pins, 
and />, these pins were in line with B, The mark on the pro- 
tractor at G was 62°. The board being held firmly in place, the 
arm was swung around until the pins at and at C were sighted 
into line with A and the reading on the protractor at H was 114°. 
How many degrees were there in the angle EOFt 

2. The distances from A and from B to the window at were 
measured and found to be 484' and 520'. A triangle O'ef (shown 
in Fig. 234) was drawn having angle eO'/=62°, and 0'e = 4.84", 
O'/=6.20". The side e/ was then drawn, measured, and found 
to be 5.18". How long is the line ^^? 

3. A woodman desires to fell only 
such trees as will funiish two 10-ft. 
cuts between the stump and the first 
limb. He wishes to allow 4 ft. for 
height of stump and waste in cutting 
at the bottom and top ends. To test 
a standing tree, he places a 4-ft. 
stick QR vertically in the ground 33 ft. 
from the tree, lies down on his back 
with his feet against the stake, and 
sights over the top of the stake to the 

first limb B, The distance from his eye P to the soles of his 

feet Q being h\ feet, should he fell the tree? 

4. In Fig. 236, if P^ = 6 ft., 72^ = 41 ft., and RA = 60 ft., how 
long is ^i?? 

5. How long is AB if PQ and RQ are the length, as in 
problem 4, and RA = 90 feet? 

6. A woodman steps off a distance of 9 steps from a tree, faces 
the tree, and holds his axe-handle at arm's length in front of his 
eye, as the ruler is held in Fig. 125, p. 189. His arm is 28 in. 
long, and he finds that 2 ft. of the axe-handle just cover the dis- 
tance from the ground to the first limb of the tree. If his step is 
3 ft. long, how high is the first limb? 




FIGUBE 236 



COMMON FRACTIONS 



331 



7. If.4^ = 35rd. 
KD (Fig. 237)? 



^ A'= 80 rd., and BV =^ 56 rd., how long is 




8. Make measures on objects in your vicinity and solve such 
problems as are suggested by those given. 

9. How long is x {Fig. 238)? 

10. The thumb is held 2 in. ^1 ""^■"-:rr=x=:^:^> 
from the end of the pencil, and the L- ."f 

pencil is held 2 ft. in front of the figure 238 

eye and parallel to the line to be" measured. When the end of 

the pencil is sighted into line with the corner i of the table, the 



^ 



FlOUBB 239 



Table 




end of the thumb is in line with tha corner a. How long is the 

end, ab, of the table, if the eye is 36 ft. from the table? 

11. The eye sights past a 
point A over the point P on 
the fence (Fig. 240), and sees 
the upper edge of the moon in 
line with A and P, Then 
moving the eye 2^ in. up the 
stake, the lower edge of the 
moon is in line with B and P, 
If the stake is 20 ft. from the 

fence and the moon is 240,000 miles away, how long is the 

moon's diameter in miles? 

12, Make measures like these yourself. 




Figure 240 



332 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



13. The rays of light from the sun passing through a pin-hole 
in the screen at H ,(F^S' ^^^) give a small circular image of the 




FIGUBB 241 



sun at /. With distances and diameter of image as shown in 
the cut, what is the diameter of the sun? 

14. A small hole, H^ in a window screen, gave a round image 
of the sun, 1.1 in. in diameter on a sheet of paper held 10 ft. from 
the pin-hole. If the sun is 93,000,000 mi. away, what is the 
sun's diameter? 

15. Fig. 242 shows a crude apparatus for finding distances that 
cannot bo measured directly. It consists of a board about 25" 
long and G" wide with end pieces about 6" high. The wood is cut 




Figure 242 

iway from the end pieces so that the eye may sight through 
between the straight edges of two visiting cards at B^ past two 
parallel threads at T, set carefully ^" apart and 25" in front of £. 

• If S is in line with the upper thread of the instrument and D 
with the lower thread, how far is it from the stake at SD to the 
instrument, if SD = 8'6"? 

SG 

Suggestion. — J : 25 = 8i : x. Then, ' Ja? = 8i X 25, or -g = 212J. 
Find X. 

16. A pupil sighted through J5' at a stake held by another pupil 
at SD' When the pupil at E sighted over the top thread, the 



USES OF SIMILAR TRIANGLES 



333 



pupil had to put his hand at S to line in with the slit at B and 
upper thread at T. The pupil at E then beckoned him to slide 
his hand down the pole until it lined in with the lower thread 
at T, The distance SB was 7.52'. IIow far was it from the 
instrument to the flagpole SB? 

17. The distance from F to the bottom of the building, Fig. 243 




Figure 243 

is 288', the distance FB = 8' and AB = 18"; how high is the 
building? 

18. In Fig. 244 it is desired to find the distance across the lake 
from C to B. A surveyor stuck a flagpole at a place from which 
he could see both and B. He measured the distances from B 
and from C to the pole he is holding, and found them to be 4563' 




Figure 244 



and 5481' respectively. He then set a pole at B, j\ of the dis- 
tance from the first pole to By and another pole at Ay -j^q of the 
distance from the first pole to C. The distance from A to B was 
measured and found to be 84.64'. How far is it from to B? 



334 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



19. It is desired to make a scale drawing of the tract of ground 
ABCDEF (Fig. 245). The parts of the apparatus to be used, as 
shown at the right, are a square board about 16"xl6", with a 




B 



m 






FiGUBE 245 

block containing a 1" hole to fit over the top pin of a sharp 
stake to be stuck in the ground to support the board. A small 
level (which may bo a phial of water) and a foot rule with a vertical 
pin at each end for sights to be used for sighting complete the 
apparatus. 

The board is set up in the field as shown, a sheet of paper 
is pinned on it with thumb tacks, and the foot rule is placed upon 
the paper. A third pin is stuck near the center of the board at a 
point (not shown in Fig. 245). Holding the edge of the foot 
rule against this center pin, sighting along the pins toward a pole 
at A the observer turns the front of the foot rule until the two 
pins of the ruler are in line with A. He holds the ruler and 
draws a line along its edge on the paper. 

He now holds the edge of the ruler against the center pin, 
and carefully sights the two ruler pins into line with B^ and, hold- 
ing the ruler in place, draws a line on the paper toward B. He 
proceeds in the same way with each of the points C, D^ E^ and 
F^ being careful not to turn the board around on the stake pin. 

20. The lines from the stake supporting the board to ^, to J5, 
to C, and so on to ^were measured and found as follows: to A^ 
460'; to B, 452'; to C, 378'; to Z>, 527'; to E, 535' and to F, 832'. 
Using a scale of 1": 100', the distance to A (460') was laid 



USES OF SIMILAR TRIANGLES 335 

off from the center pin on the line drawn toward A giving a (on 
the board), the distance to B (462') was laid off from the center 
pin on the line drawn toward B giving b (on the board), and so 
on around to F. Calling the center pin o, how long is oa? ob^ 
oc? od? oe? of? 

21. With a ruler the points a, S, c, d, e and / were connected 
as shown in the figure. The lines were ab = 2.8"; be = 4.5"; cd = 
5.12"; de = 2.95"; 6/= 6.25"; fa = 4.48". How long are the lines 
AB, BC, CD, DE, EF, and FA? 

The board may be supported by a light camera tripod or by a 
home-made tripod and the board may be held to the flat top of the 
tripod by a thumb nut. (See Fig 246.) 

22. The distances from the apparatus to the corners of the field 
(Fig. 246) were: to J, 678'; to B, 612'; to (7, 683'; to D, 738'; 



..^^mm--^^ 



Figure 246 

to E, 698'; to F, 625'; to G, 679'. Using a scale of 1":200', 
how long should the distances be made from the center pin to a? 
to b? to c? to d? to e? to/? to g? 

23. With a ruler a, 5, c, c?, e, /, g, and a were then connected 
and lines measured. The measures were: ab= 3.81"; Sc = 4.12"; 
ceZ = 2.86"; e?c = 2.75"; e/=5.86"; /(7 = 6.18" and ^fl = 5.98". 
How long are the lines AB? BC? CD? DE? EF? FG? GA? 



33G RATIONAL GRAMMAR SCHOOL ARITHMETIC 

24. Having made the scale drawings, perpendiculars may be 
drawn from the center pin to each of the pides ab^ J^ and so on. 
From the measured lengths of these perpendiculars and the bases 
AB, BC^ and so on of the triangular parts of the figures, the areas 
of the triangles aoh^ boc, and so on (calling o the point where the 
center pin stands) may be computed. The sum of these areas 
gives the total area of the figure abcdefga, 

25. Suppose the area of triangle aob were 4.94 sq. in. what 
would be the area of A OB ( being the point on the ground just 
under o), if the scale were 1":200'? (See Problems 13 and 15, 
§190). 

26. If the areas of the triangles aoby boc^ cod^ doe^ eof^ and 
foa (Fig. 245 prob. 7) in square inches were: 6.073, 7.740, 9.172, 
7.490, 16.725 and 7.563, respectively, and the scale were 1": 100'; 
what were the areas of the triangles AOB^ BOC^ COD^ DOE^ 
EOF, and FOA'^ 



AFFLIGATIONS OF FERGENTAOE 

§192. Insurance. 

1. What will it cost to insure $680 worth of household furni- 
ture, at the rate of If %? 

Definition. — The amount paid for insurance is called the premium. 

2. An art gallery, valued at $500,000, is insured at the rate of 
li%. What is the premium? 

Definitions. — The written agreement between an insurance company 
and the insured is called a policy. 

The amount for which the property is insured is the /ace of the policy. 

3. A vessel, valued at $16,000, was insured for f of its value. 
Find the face of the policy. 

4. A growing crop was insured at 5%. ' The premium was 
$140. What was the face of the policy? 

5. If it costs $420 to insure a house for J of its value, at 3.J^%, 
what is the house worth? 



APPLICATIONS OF PERCEXTAGE 337 

6. A stock of hardware was insured for 17000, insurance cost- 
ing 1110.75. What was the rate ot insurance? 

7. If a shipment of grain is worth 1840, and the premium 
amounts to $17.60, what is the rate of insurance? 

8. A machinist insured his tools, valued at $360, for i of their 
value, .paying $8.19. What rate did he pay? 

9. A farmer paid a premium of $10.50 for insuring his stock 
at 1^%. For what amount was the stock insured? 

10. A man paid $67.50 for the insurance of a steam launch at 
1^%. For what amount was the launch insured? 

11. What is the rate paid for insuring a bridge, valued at 
$15,000, for f of its value, the premium being $240? 

12. A library, worth $28,000, was insured for f of its value, 
the premium being $720. What was the rate of insurance? 

13. The contents of a grain elevator were insured at a rate 
of f %. What was the amount of the policy, the premium paid 
being $272.40? 

14. The premium paid for insuring a quantity of lumber, for 
two-thirds of its entire value, at 3 % , was $36. What was the value 
of the lumber? 

15. A piece of property, valued at $27,200, was insured 
fori of its value. The premium was $212.50. What was the 
rate? 

16. A tank of oil, holding 2592 gal., worth 18^ per gallon, 
was insured at 4%. Find the premium. 

17. A man insured a cargo worth $2400 at 3^%. f of the 
cargo was lost at sea. Wliat amount of insurance should the 
man obtain? What premium had he paid? 

18. What is the rate of insurance paid for insuring 26,000 bu. 
of grain Vorth 77^, for f of its value, if the premium is $500.50? 

19. What sums must be paid for policies to insure a factory, 
valued at $21,000, at If % ; a house, worth $28,000, at i% ; and a 
barn, valued at $5600, at If %? 

20. A steamboat had a cargo, valued at $2000, which was 
destroyed by fire. The insurance was $1500. What per cent of 
the value of the cargo was covered by insurance? What was the 
premium, at 4^%? 



338 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

§193. Taxes. 

Definition. — A tax is a sum of money levied by the proper officers 
to defray the expenses of national, state, county, and city governments, 
and for public schools and public improvements. 

1. A town wishes t.o raise 123,500 to build a public hall. If 
the collector's commission is 2|%, what is the total amount to be 
collected? 

2. If the taxable property of a town is valued at $923,846 
and the rate of taxation is 3f %, what is the whole amount of 
the tax? ' 

3. The taxable property of a town is $869,472, and the rate 
is 8^ mills on the dollar. What is the amount of the tax? 

4. What amount of money must be collected to raise a net tax 
of $643,000, allowing 6% for collection? 

5. The tax in a city is $49,682; the rate is 15 mills on the 
dollar. What is the assessed valuation? 

Definition — Assessed valuation means the estimated value of the 
property that is assessed. 

6. What IS the rate of taxation when property assessed at $8940 
pays a tax of $160.92? 

7. The net amount collected as a. tax was $2896.42. The 
collector's commission was 3^%. What was the whole amount 
collected? 

8. The assessed valuation of the taxable property of a town 
was $1,218,694. The tax to be raised was $18,280.41. How 
many mills on the dollar was the rate of taxation? 

9. A tax collector received $175.18 as his 2% commission on a 
certain sum collected. The assessed valuation of the taxable 
property was $922,000. What was the tax of a man, whose prop- 
erty was valued at 118,000? 

10. The real property (houses, lands, etc.) of a certain town 
is valued at $4,560,800, and the personal property (movable 
property) at $945,900. If $12,000 is to be raised by taxation, 
how much must a man pay, whose property is valued at 
$9,840? 

11. A net tax of $8965 is to be raised in a certain city. The 



APPLICATIOXS OF PERCEKTAOE 339 

assessed valuation of the taxable property is $5,689,243. The 
collector's commission is 1^%. What will be the tax of a man, 
whose property is valued at 156,000? 

12. Property on a city street is assessed 2% on its valuation. 
How much more will a man pay who owns 100 front feet, valued 
at $125 per foot, than one who owns property valued at 
$7500? 

13. If the assessed valuation of the taxable property in a cer- 
tain city is $1,069,210, and the whole amount of tax collected is 
$13,899.73, what is the rate of taxation and the net amount after 
deducting the collector's commission of 1|%? 

14. The taxable property in a town is $872,990. The rate of 
taxation is .015. What is the net amount collected after deduct- 
ing the collector's commission of 2^%? 

15. The assessed valuation of the taxable property in a certain 
city is $4,968,390. The tax collected amounts to $94,399.41. 
How many mills on the dollar was the rate of taxation? 

16. A certain town wishes to raise by taxes $20,500 to build 
a schoolhouse. What tax must be levied to cover this and the 
cost of collection at 4%? 

§194. Trade Discount. 

1. A hardware dealer bought the goods named in the follow- 
ing bill : 

(1) 2 doz. braces, @ 45^ each; 

(2) 50 1b. J" bolts, @ 3i^; 

(3) 12 bales barb wire, @ $2.15; 

(4) 25 kegs nails, @ $2.50; 

(5) 8 cooking stoves, @ $28.75; 

(6) 12 baseburners, @ $32.50; 

(7) 150 lb. screws, @ $4.75 per cwt. 

The bill was subject to discounts of 10%, 7%, and also 5% 30 
da. or 6% 10 da. What was the total cost if the bill was paid in 
30 da.? in 10 days?, 

2. Find the amount needed to settle the following bill, sub- 



340 . RATIONAL GRAMMAR SCHOOL ABITHMETIC 

ject to the successive discounts, 10%, 8%, and 5% off for 30 da. 
or C% for cash, the bill being paid in cash: 

(1) 2 chests carpenter's tools, @ $48.50; 

(2) 2 doz. augurs, @ 25^ each; 

(3) 2 doz. carpenter's rules, @ $1.80; 

(4) 480 lb. hinges, @ $4.75 per C; 

(5) 3 doz. tin buckets, @ 22^ each; 

(6) 4 doz. gardening rakes, @ 28^ each; 

(7) 14 scoop shovels, @ 85^; 

(8) 6 wash boilers, @ 78^; 

(9) 12 washtubs, @ 52^; 

(10) 10 doz. clotheslines, @ $1.10. 

3. What amount will be needed to settle the bill of problem 2 
in 30 days? 

4. What amount will settle the following bill of house furnish- 
ings, the bill being subject to the discounts, 20%, 10%, 8%, and 
5% 30 da., 6% cash, the bill being paid in 30 da.? 

(1) 8 dining tables, @ $30; 

(2) 12 sets dining-room chairs, @ $18.50 a set; 

(3) 10 bookcases, @ $12.50; 

(4) 8 rocking chairs, @ $12.75; 

(5) 15 center tables, @ $15; 

(6) 12 Wilton rugs, @ $25; 

(7) 15 Axminster rugs, @ $23; 

(8) 250 yd. ingrain carpet, @ 45^. 

5. What amount paid in cash will be needed to settle the bill? 

6. What amount in cash will settle this bill of plumber's 
supplies : 

(1) 40' lead pipe, @ 15^, discounts 20%, 10%, 7%, 2 f^ 

cash; 

(2) 100' iron tubing, @ 3^^, discounts 20%, 10%, 7%, 

2% cash; 

(3) 8 bathtubs, @ $20, discount's 20%, 7%, 2% cash; 

(4) 8 lavatory outfits, @ $10, discounts 20%, 7%, 2% 

cash, 

(5) water meters, @ $5, discounts 20%, 12%, 2% cash; 
{()) 4 pumps, @ $25, discounts 20%, 7%, 2% cash; 

(7) 5 pump valves, @ $2.85, discounts 20%, 7%, 2% 
cash? 



APPLICATIONS OF PERCENTAGE 



341 



§195. Stocks and Bonds. 

A Stock is a written agreement (called also a stock cerlificate)^ 
made by a company to pay the holder a certain part of the eara- 
ings of the company. The sum paid is reckoned at a certain 
number of dollars per share^ or at a certain rate per cent of the 
face value of a share. A share usually has a face value of $100, 
$500, $50, $1000, etc. A share of mining stock often has a much 
smaller face value. 

When a stock company pays to the holders of its stock $2 on 
every $100 of its capital stock, or 2% on its stock, the company is 
said to be paying a S^ dividejid^ or a ^% dividend. 

When stock is paying a high rate of dividend it may sell in the 
market above par^ or for more than its face value. If the rate of 
dividend is low, the stock may sell for less than its face value, or 
below par. When it sells for its face value it is sold at par. 
city government, or by a company, to pay the holder a stated sum 
of money with interest on it at a stated rate. The sum of money 
to be repaid is called the face of the bond. 

Stocks and Bonds are usually purchased tHrough an agent, 
called a broker^ who makes a business of buying and selling stocks 
and bonds. He charges a certain per cent (usually i%) of the 
face value for buying, and an equal rate for selling. This charge 
is called brokerage. 

In the following problems, regard the face value of the stock 
or bond as $100 and the brokerage as ^%, unless otherwise stated. 

Following is a list of newspaper quotations on the N. Y. stock 
market on certain stocks and bonds : 



STOCKS 



Open 

Amal. Copper. 9lb% 

American Sugar 129^ 

B. &0.com 101^ 

B. &0. pfd 9554 

Chi. & Alton com. . . . 36J4 

Chi. & Alton pfd 71 Vi 

111. Central R. R 148 

Peoples Gas \Qlb% 

Pressed Steel com . . . 65Vi 
Pi-essed Steel pfd. . . . 94Vi 
Rubber Goods com.. 24 V4 

U.S. Steel com ^\ 

U. S. Steel T)fd 87% 

Si. L.. and S. F. com. 78H 
St. L. and S. F. pfd. . 79% 



No. Sales 
27000 Atchison Gen. 4s. 
12000 B. and O. gold 4s.. , 

1000 B. andCSHs 

1000 C. B. andQ. 4s 

POOO'C. and A. S^s 

28000 U.P Is 



HiRh Low Noon 
66Ji 9ib% es>% 
130 129^ 129S1£ 
101^ 1005^ 101 
95^ ^5Vi 95H 
37 86^ 36^ 
72^ 71>i 72 
148^ 148 148^ 
105% 105^ 105% 
eSH 66J^ 651/3 
941/a 9414 94H 
2A% 241/, 241/s 
m\ 37H 37H 
87H 81^^ 87M 
90% 78 80J^ 
80/8 rdJi 80J/a 



BONDS 

Closing bid and asked prices for govern- 
ment bonds were as follows: 

nid Asked 

New 2s 1 08 -^^ 1 09 14 

Coupons It 8^ 109J^i 

New 3s IO8M 109 

New 3s coupon 108?^ 109 

New 8s small 108 109 

Registered 4s 112^ 113 

Coupon 4s . . 112H 113 

Registered 4s new 139J^ 1395^ 

Coupon 4s new 1393i 139% 

Registered 5s 107^ \m% 

Coupon 58 107J4 10796 



BOND TRANSACTIONS 



.100H@100^ 
.101 @101H 
. 94H 
. 106^^ 

76H@ 76?^ 
.102>^ 



No. Sales 

5000 U. S. Leather 4s 85J^ 

3000 B. «& O. coupon 4s 106M@1063g 

82000 Cen. Pac. R. R. 8Hs 88k <S) 8«?% 

10000 Wabash R. R 5.s 11r>k(^115i5 

ICOOO Manhattan 4s 105^ 



343 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

1. A certain express company has a capital stock of $500,000, 
divided among its stockholders in shares of the par value of $100 
each. In 6 mo. the net profits amount to $10,500, which the 
company distributes among its stockholders as a dividend. What 
is the rate of dividend? How much does a holder of 200 shares 
receive as dividend ? 

2. What must a man have paid for 400 shares AmaL Copper 
stock on the date of the table, including brokerage, if he bought 
at the opening price? at the highest price? at the lowest? at the 
noon price? 

3. Answer similar questions for U. S. Steel pfd. (preferred). 

Definitions. — Preferred stocks are stocks which pay a fixed dividend 
(say of 7%) before any dividends are paid on common stocks; Common 
stocks pay dividends dependent on the net earnings of the company 
after expenses and dividends on preferred stock have been paid. 

4. How much did a man gain or lose by buying 2000 Am. Sugar 
at the opening price and . selling at the highest price, paying 
brokerage of i% for buying and also for selling? 

5. Answer similar questions for B. & 0. com. ; for B. & 0. pfd. 

6. How much does a man make or lose who buys 2500 shares 
111. Cen. R. E. stock at the lowest quotation of the table and 
sells at the highest, paying brokerage of i for buying and ^ for 
selling? 

7. If a man invests in St. L. and S. F. pfd. stock, paying 7% 

annual dividend, at the lowest quotation of the table, what interest 

does he receive on his investment? 

Suggestion.— The investor must pay |79| + IJ per" share (of $100), 
and he receives ^7 per year as interest. 

8. Answer a similar question on C. & A. pfd. 

9. What interest does a man receive on an investment in Gov- 
ernment Xew 2s if he invests at the price "bid," brokerage ^? at 
the price "asked"? 

Note. — Government 2s, 3s etc., are government bonds paying 2% 
3fo, etc., annually. 

10. Answer similar questions for New Ss; for New 3s cou- 
pon ; for Eegistered 4s ; for Registered 4s new. 

11. What rate of interest does the investor who bought the 
1000 B, & 0. 3^8 receive? 



APPLICATIONS OF PERCENTAGE 343 

12. Answer similar questions for the investor who bought the 
9000 C. & A. 3^8, if he paid the lowest quoted price; the highest. 

] 3. Which pays the higher rate of interest on the investment, 
and by how much, U. P. 4s as quoted, or Wabash R. R. 5s at 
ihe lowest quotation? at the highest? 

14. Which is the more profitable investment, and by how 
much, B. & 0. coupon 4'8 at the lowest quotation, or a straight 
loan at 4%? 

15. Answer other similar questions on the table. 

16. An investor in Chi. & Alton com. stock at the highest 
quotation received two 2% and one 3% dividend, and sold the 
stock 18 mo. later at 38^. What rate of interest did ho receive 
on his investment? How much profit did he make if he bought. 
1000 shares? 

17. Answer similar questions on the table. 

§196. Compound Interest. 

With certain classes of notes, if the interest is not paid when 
due, it is added to the principal and this amount becomes a new 
principal, which draws interest. Savings banks add the interest 
on savings deposits at each interest-paying period and pay interest 
on the entire amount. This interest on interest is called com- 
pound interest. Bankers also collect their interest on loans and 
then reloan the interest, and in this way virtually receive com- 
pound interest on their money. 

Compound interest is usually payable annually, or semi- 
annually. 

1. Find the compound interest on a note of $200 at 6% for 
5 yr., interest payable annually. 

Solution.— 

Interest on 1200 for 1 yr. at 6% = 200 X $ .06 = % 12.00 
Amount of $200 for 1 yr. at 6% = 200 X S 1.06 = $212.00 
Amount of $212 for 1 yr. at 6% = 200 X $(1.06)2 = §224.72 
Amount of $224.72 for 1 yr. at 6% = 200 X $(106)3 = $238.23 
Amount of $238.23 for 1 yr. at 6% = 200 x $(1.06)* = $252.52 
Amount of $252.52 for 1 yr. at 6% = 200 X $(1.06)^ = $267.68 

The compound interest=$267. 68— $200=$67. 68. 

Note. — The $267.68 is called the compound amount The expressions 
(1.06)', (1.06)*. and (1.06)6 mean 1.06 X 1.06 X 1.06, 1.06 X 106 X 106 X 
1.06, and 1.06 X 1.06 X 1.06 X 106 X 106. and are read ''1.06 cube," 
**1.06 to the 4th power," and "1.06 to the 5th power." 



344 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

2. Find the simple interest on a note for the same face at the 
same rate and for the same time as in problem 1. By how mnch 
does the compound interest exceed the simple interest? 

3. Find the compound interest on $180 at 6% for 3 yr., interest 
compounded semi-annually. 

Solution.— 

Amount on |180 for J yr. at 6% = 180 X 1 1.03 = $185.40 
Amount on $185.40 for J yr. at 6% = 180 X 1(1.03)2 =^ $i90.96 
Amount on 1190.96 for J yr. at 6% = 180 X $(1.03)' = $196.69 
Amount on $196.69 for J yr. at 6% = 180 X $(1.03)* = $202.59 
Amount on $202.59 for i yr. at 6% = 180 X $(1.03)^ = $208.67 
Amount on $208.67 for J yr. at 6% = 180 X $(1.03)» = $214.93 

Compound interest = $214.93 — $180 = $34.93. 

4. If P denotes any principal at r% for n yr., interest being 
compounded annually, show that if A denotes the compound 
amount, 



-( 



ry 



1 + 100/ 

5. Show that if the interest is compounded semi-annually, 

the letters meaning the same as in Problem 4. 

6. Show^ that if / denotes the interest, compounded annually, 

7. If the interest is compounded semi-annually, show that 

8. A man has a deposit of $100 in a savings bank paying 4%, 
interest compounded semi-annually, for 3 yr. How much is due 
tlie depositor at the end of that time? 

0. How much is due on a savings deposit of $250, in a savings 
V)ank paying 3% semi-annually, the deposit having been in the 
bank 3^ years? 



USB OF LETTERS TO REPRESENT NUMBERS 345 

10. A man paid compound interest at 6% , interest compounded 
annually, on a note of $150 for 7 yr. How much was required to 
pay the note? 

Note. — Many notes have coupon notes attached for the amount of 
interest due at each interest-paying period. These coupon notes usually 
bear a higher rate of interest than does the note itself. 

11. A note for $500, bearing 6% interest, has 3 interest 
coupons attached, the coupons, when due, bearing 7% interest, 
payable annually. If the note runs 4 yr., no coupons being paid 
in the meantime, how much money will be required to pay the 
entire debt at the end of this time? 

12. A note of $450, bearing interest at 7%, interest pay- 
able semi-annually, coupons bearing 8% interest, amounts to 
what sum at the end of 4^ yr., no coupons being paid until the 
end of the time? 



USE OF LETTERS TO REPRESENT NUMBERS 

§197. Problems. 

1. Joseph had 45^ and he earned 15^ more selling oranges. 
IIow many cents had he then? (Answer by indicating the opera- 
tion you perform, thus: 45^ -f- 15^.) 

2. William had x marbles and Harold gave him y marbles. 
How many marbles had he then? 

3. A cow gave x lb. of milk at the morning milking and y lb. 
at the evening milking. How many pounds did she give at both? 

4. James earned 80 jJ selling papers on Saturday and spent 45^ 
of his earnings. How many cents did he save on Saturday? 
(Answer by indicating the operation you use.) 

5. During July a boy earned m dollars and spent s dollars. 
How many dollars did he save? 

6. Helen bought 15 pencils at 3^ apiece. How much did she 
pay for all? 

7. Elizabeth took x music lessons during February at y dollars 
per lesson. What was the cost of her lessons for the month? 



346 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

8. A thermometer rose ^° on one day and 5 times as much 
the day following. How much did it rise on the following 
day? 

9. The area of a rectangular lot is 63 sq. rd. and the length of 
one side is 9 rd. How long is the other side? 

10. The area of a rectangle is x square inches and the length 
of one side is \j inches. How long is the other side? 

11. How many lots each of h ft. frontage can be made from a 
frontage of a feet? 

12. An orange boy earns a cents on Wednesday, three times 
as many cents on Thursday, and as many cents on Friday as on 
Wednesday and Thursday together. On all three days he earns 
80^. How. much does he earn on Thursday? on Friday? 

Note. — In all such problems use the equation. We have a + 3a -|- 4a 
= 80, or 8rt = 80. If 8a = 80, a = 10. 3a = 30, Thursday earnings, and 4a 
= 40, Friday earnings. 

13. The altitude of a rectangle is a; in. and the base is Zx in. 
How long is the perimeter? What is the area of the rectangle? 

14. 4a means 4 x a. Compare the values of a x 4 x a and 4 x 
a X a. How is a X a written? How is 4 x a x a written? Ans, 4a". 

15. The mercury stood at x degrees at 2 p.m. and fell 3° dur- 
ing the next hour. The reading at 3 p.m. was 25°. What was 
the reading at 2 p.m.? 

16. The mercury stood at 12° at 8 a.m. During the next two 
hours it rose x degrees. What was the thermometer reading at 
10 a.m.? If it had fallen y degrees, what would have been the 
reading at 10 a.m.? 

17. The thermometer read 28°, the mercury fell x degrees, 
then Zx degrees, and then rose ^x degrees, when the reading was 
32°. What was the number of degrees in each o? the three 
changes? 

18. James paid %x for a hat and twice as much for a coat. 
He paid $4.50 for both. What did he pay for each? 

19. I paid 130 for a bicycle and sold it for %x. How much 
did I gain? 

20. I paid ^ of what I gained for a coat. How much did I pay 
for the coat? 



USB OP LETTERS TO REPRESENT NUMBERS 347 

21. Louis had x apples and aj^e y of them. How many did he 
have left? 

22. Henry had 7a papers and sold 4a of them. How many 
did he have left? 

23. I walked x miles due south one day and y miles duo north 
the next day. How far was I then from my starting point? 

24. A man rows a boat downstream at a rate that would carry 
his boat a miles per hour through still water, and the current 
alone would carry him down h miles per hour. How far will he 
go in 1 hour? 

25. A man walks from rear to front through a railway coach 
3 mi. per hour, and the coach is running at the same time a mi. 
per hour. How fast does the man pass the telegraph poles along 
the track? 

26. How fast does he pass them if he walks from front to 
rear? 

27. Mary had a pencils and sold them at 5^ apiece. How 
many cents did she receive for them? 

28. James sold x oranges at a cents apiece. How many cents 
did he receive for them? 

29. A dealer sold a wagons for 40a dollars. What was the 
price per wagon? 

30. A farmer paid x dollars for 15 A. of land. How much 
did he pay per acre? 

31. The area of a rectangle is m sq. rd. and it is I rd. long. 
How wide is the rectangle? 

32. A township is x mi. square and contains 36 sq. mi. What 
is the value of a:? What does x represent? 

33. Helen bought x dolls at 10^ apiece and y yd. of muslin at 
8^ a yard. How many cents did she pay for both? 

34. James had m cents and earned c cents more. He invested 
all his money in papers at 3^ apiece. How many papers did 
he buy? 

35. He sold his papers at 5^ apiece. How much did he receive 
for the papers? How much did he gain? 

36. The base of a triangle is x ft. and the altitude is y ft. 
What is the area in square inches? 



348 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

37. The area of a triangle is a; sq. ft. and the base is 6 ft. long. 
What is the altitude? 

38. A parallelogram has a base {x + y) in. long, and is 6 in. high. 
What is the area? Express this answer in two ways and make an 
equation by writing the two expressions equal. Why are the two 
expressions equal? 

§198. Uses of the Equation. 

1. If each brick weighs 7 lb. and there are 
10 bricks in the bucket (Fig. 247), with how many 
pounds of force does the bucket pull downward 
on the rope, the bucket itself weighing 5 pounds? 

2. If we denote the number of pounds of force 
with which the man pulls downward on the rope 
to balance the bucket by p, write an equation 
showing the number of pounds in p, the 5-lb. 
bucket being loaded with 10 brickg. 

3. A horse is raising a ^ 
10-ft. steel I-beam weighing 

FiGURB 247 30 lb. for each foot of length. 
If the force the horse must exert to hold the 
beam suspended in the air be denoted by JP, 
write an equation showing the number of 
pounds in F. 

4. A wagon weighing 1800 lb. is loaded 
with 3 T. of coal. When it is being drawn ^^''''^^ ^^® 
over ordinary pavement it pulls backward on tbe traces of the 
team with a force of ^V ^^ many pounds as there are pounds in 
the entire weight of the coal and wagon. If the force exerted by 
the moving team is F lb. , write an equation showing the number 
of pounds in F. 

5. Write an equation showing how many pounds each horse 
draws if both draw with equal force. 

These probbms show how the equation may be used in solving 
simple problems of mechanics; and we need to learn the laws upon 
which the use of the equation is based. 





USE OP LETTERS TO REPRE8EKT NUMBERS 349 

§199. Principles for Using the Equation. 
Beview p. 97, 

1. Two weights, one of x lb. and the other of y lb., are tied 
to strings which pass over pulleys at A and B 
(Fig. 249). The strings are knotted, at C. If x 
is greater than y, how will C move? Under what 
condition will G remain at rest? What relation 
is shown to exist between x and ^ by a move- 
ment of C toward the right? 

(T's standing stationary indicates a balance %n value between 
the forces y lb., drawing toward the right, and x lb., drawing 
toward the left. This fact is expressed in symbols by writing 
a: lb. = ^ lb., or, better, by a: = y, simply. 

2. If a 10-lb. weight be hung to the weight y, how many 5-lb. 
weights must be hung to a; to secure a balance of value 9 

Note. — ^This is shown by writing y+10s=a?+5 + 5, read **y plus 10 
equals X plus 5 plus 5." 

3. How many pounds in all were added to xl 

4. If 50 lb. were added on the right, how many 5-lb. weights 
must be added on the left before the sign of equality (=) can be 
written between the numbers? If 7a lb. were added on the right, 
how many a-lb. weights would be needed on the left for balance? 

5. How many pounds would be needed on the right if b lb. are 
added on the left? S + c lb.? jt? + 35 pounds? 

6. Write the equation form of statement for each case of Prob- 
lems 4 and 5. 

7. If a greater number of pounds were added to y lb. than to 
X lb., say 15 lb. to y lb. and 10 lb. to x lb., what would occur in 
the apparatus shown in Fig. 249? 

This destroying the balance, or emmlity. Is stated briefly in signs by 
writing y + 15 > X + 10, which is read *'y plus 15 ia greater thanx plus 10." 

If the balance, or equality, is destroyed by adding a heavier weight to 
X than to y, the fact is stated in symbols thus: j^ + 10 < cc + 15; read 
**y plus 10 is less tJian x plus 15." 

Notice in each case that the vertex at the horizontal V always points 
toward the smaller number. 

Definitions. — An expression in which the sign < or > stands between 
two numbers is called an expression of inequality. 



350 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

We now have signs for writing briefly the three possible relations 
which may exist between any two numbers, as a and b. They are called 
relation signs. 

8. Read and give the meanings of (1) a > J, (2) a = S, (3) 
« < S, (4) ic + 20 < a; + 15 -f 10. 

9. It y = x^ and a lb. are added to y lb. and b lb. to x lb., how 
will the knot C move for case (1) Problem 8? for case (2)? for 
case (3) ? State in symbols the fact shown by the knot C in each 
case after adding the weights, using the correct relation signs. 

10. If any given weight, a lb., be added to ^, what must be 
true of the weight b lb. if, when added to a;, we may write 
y-^a = x + b? 

11. If z lb. be added to both the y lb. and the x lb., when 
y = a:, what equation may we write? 

Principle I (For Addition of Equations). — 1/ the same 
number y or equal numbers^ be added to both sides of an equatioii^ 
the sums' are equal. 

Illustration. — If y=^x, and a = 6, then we may write 



and y 



-j- a := x - - a, 
b = x-j-b. 



and y + a = x + 6, 
and y + 6 = a? + «. 

12. If 8 lb. be removed from the left scale pan (Fig. 249), how 

many 4-lb. weights must be removed from the right pan to restore 

the balance ? 

Note. — This is written in signs y — 8 = a? — 4 — 4, and is read *'y — 8 
equals x minus 4 minus 4." 

13. How many pounds in all were removed from the right 
scale pan? 

14. It y = Xj and if a lb. be removed from the left, and b from 
the right, to what must the value of b be equal to enable us to 
write y — a = x — b? 

Principle II (For the Subtraction of Equations). — If the 
same number^ or equal numbers, are subtracted from equal num- 
bers, the differences are equal. 

Illustration. — If y = a? and a = 6, then we may write: 

y — a = x — a, 
and y — b = x — b, 
and y — a = x — 6, 
and y — b = x — a. 



USE OF LETTERS TO REPRESENT NUMBERS 351 

15. If in Fig. 249 y be doubled, what corresponding change in 
X will restore the balance? 

Note. — The double of y is written 2y and read "two y." 

16. What equation states that there is a balantje? 

17. li y = x^ and if 4 weights, each equal to y, are put on the 
right of the apparatus in Fig. 249, how many weights, each equal 
to x^ must be put on the left to keep the balance? 

18. If a weights, each equal to y, are on the right, how many 
weights, each equal to x^ must go on the left to secure balance? 

Note.— The equation is ay = ax, 

19. If a = S, and a weights, each equal to ?/, are put on the 
right, and b weights, each equal to re, are put on the left, what 
will be shown by the scales if y = x. (Answer with an equation,) 

Principle III (For the Multiplication of Equations). — 
If equal riumbers are multiplied by the same number^ or by equal 
numbers^ the products are equal. 

Illustration. —If y=^x and a = 6, then 

ay = ajx, 
and by = bx, 
and ay = bx, 
and ax = hy. 

20. If y = Xy and half of the weight on the left be taken off, 
what fractional part of the weight on the right must be taken off 
to restore the balance? 

21. If only ^ of y be kept on the right (Fig. 249), what frac- 
tional part of X must remain on the left? 

22. If only ^ lb. be kept on the right (Fig. 249), what frac- 
tional part of X lb. must remain on the left for balance? 

23. Suppose y = X and a = b, and that — lb. are on the right, 

a 

X 

and -T lb. are on the left ; what equation would be shown to be 
true by the apparatus? 

Note. — The equation is — =-r, read "y divided by a equals x divided 
a 

by 0." 



352 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

Principle IV (For the Division of Equations). — If equal 
numbers are divided by the same 7iunibery or by equal numbers^ 
the quotie?its are equal. 

Illustration.— If y = x and a = 6, then 





y_x 
a a* 


and 


y_x 
6" 6' 


and 


y_x 


and 


y_x 
b a' 



In Fig. 183, p. 293, a {x-\-y) represents the area of the whole 
rectangle, while ax and ay represent the areas of its two parts. 
As the two parts of the large rectangle, taken together, must 
equal the whole rectangle, we may write : 

a{x + y) = ax-^ay 

In Fig. 184, p. 294, the area of the whole rectangle is given 
by (a + b) {x -f y) , while the areas of the several parts are ax, ay, 
bx^ and by. Since the parts, taken together, make up the whole 
rectangle, we may write: 

(a + J) (a; + y) ^ ax + ay + bx-^- by 

These two equations illustrate the meaning of a fifth principle 
of the equation, viz. : 

Principle V. — Any whole equals the sum of all its parts. 

These five fundamental principles or laws, exemplified by the 
scales, p. 97. and the pulley device (Fig. 249), p. 349, 7nust 7iot 
be violated in using the equation. 

§200. Problems. 

1. Find the value of the letter rr, y, or z, in each of the 
first nine problems: 

(1) 3a: = 15 lb. ; (4) dz = 36i^; (7) llic = 55; 

(2) Sx = 24 ft. ; (5) ix = $3 ; (8) iy = 21 ; 

(3) 7y = 18 mi. ; (6) iy = 14 sq. in. ; (9) ax = 3a. 

2. If 3a; = 9, to what is 2x equal? 7a;? |:r? 

3. If 7.5 = 21, to what is 5.c equal? far? eja;? 



USE OP LETTERS TO REPRESENT NUMBERS 353 

4. If ic + 3 = 8, what is a;? 3a;? 5a;? ic*? __ 

5. If 2a; + 7 = 15, what is a;? 3a;? 7a;? ^? s/x'i 

6. If a; - 3 = 6, what is xt 3a;? Q?t y/x? 

7. 2x 4- 3a; + 6a; = 33; find x. 

8. 4a; - 2a; + a; = 64; find a;. 

9. 7a;-a; + 2a; = 32; find a;. 

10. xy = 16, and ^ = 8 ; what is a;? If a; = 8, what is y? 

11. ax = 35, and « = 5 ; what is x? If a- = 5, what is a? 

12. ab = 10 1 what is ^ if a = 1? 2? 3? 4? 10? 20?- 

13. By what principle may we write 

c(x + y-{'z) - cx + cy + cZj 

Xj y, and z denoting the bases of 3 rectangles whose altitudes are 
each equal to f? (Answer by sketching the proper figure and 
pointing out the rectangles whose areas represent each of the 
products in the equation.) 

14. If a = 8 and & = 5, find the values of the following expres- 
sions and tell what ones are equal : 

(1) (a + h); (4) {a + by ; (7) a' - 2ah + h' ; (10) a' + ¥ ; 

(2) {a -b); (5) (a - bf ; (8) a' + 2ab + ¥ ; (11) a{a' + V) ; 

(3) ^aS; \&)a'-\-b^i {^) {a + b){a-b); {12) b{a-b). 

15. If a; = 9 and y = 4, tell which of the following express true 
relations and write the correct relation sign in each case: (See p. 
350.) 

(l)a:>y; (11) {x + yf^i^ + yy-, 

(2)a; = y; (12) {x-yf = x'-f^ 

(3) x<y', (13) {x-^yY = x^ + 2xy-\-y^i 
{'l)y<x\ ' {U) lx-yy = x^-2xy + y'^; 

(5) a; + y = 13; (15) {x-yf = x^ + 2xy + y^; 

(6) a;-y = 5; (16) {x + y){x-y)<x'-f', 

(7) (a; + ^)2 = 169; (17) {x+y){x-y)=x' -^2xyi^y''; 

(8) {x-yr = 25; (18) {x-y){x-^y) = (a^ + f); 

(9) a:2-jy«>25; (19) {x-y){x4-y) <{x^ -y^); 
(10) ar^-fy2 = 169; (20) x{x + y)^ x'-i-xy. 



354 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

§201. Statements in WorcU and in Symbols. 

1. Write the symbolic statements for these verbal phrases and 
statements. Let x stand for the number when there is but one 
number to be symbolized in the problem. 

(1) A certain number increased by 15. 

(2) Twice a number diminished by 8. 

(3) Seven times a number increased by three times the 
number. 

(4) The square of a number, divided by 8. 

(5) The sum of the square and the first power of a number. 
(G) Eight times a number, divided by three. 

(7) One-third of 10 times a number. 

(8) Three times a certain number, diminished by one, equals 20. 

(9) Eighteen times the square of a number equals 72. 

(10) Twenty-five times a number, increased by 5, equals 30 
times the number, diminished by 15. 

(11) One-eighth of the sum of a certain number and 18. 

(12) Six times the difference between a certain number and 
3 equals 18. 

(13) The product of the sum and the difference of x and 3 
equals 18 (x being greater than 3). 

(14) The difference between ic and 18 is greater than 18; is 
less than 25; is equal to 20 (x > 18 in each case). 

2. State in words what these expressions mean. For example, 
(1) means "double a certain number, diminished by 9": 



(1) 2^-9; 


(6) (a;+l)(ar + l); 


(11) 2a; + 6 = 12; 


(2) lG.r + .»-; 


(7) x{x-^y. 


(12) 7a:-2 + 16; 


(3) 28a; + 17; 


(8) 3(9-0:); 


(13) 5a:+7 = 42; 


(4) .^ + .n 


(9) 12(^-1); 


(14)(a;-4)(a; + 4)=20; 


(5) .«-.<•; 


(10) (^-l)(x-l); 


(15) {a^-V){(l-V) = a*-h*. 



3. Translate into symbols these verbal phrases and statements, 
using a and ^, or x and y, for the two numbers : 

(1) The sum of two numbers equals 25. 

(2) The difference of two numbers equals 15. 

(3) The sum of the squares of two numbers is less than 27. 

(4) The square of the sum of two numbers equals 100. 



/u 



USE OF LETTERS TO REPRESENT NUMBERS 355 



(5) The difference of the squares of two numbers equals 9. 

(6) The sum of the squares of two numbers equals seven times 
the difference of the numbers. 

(7) The product of two numbers equals their sum. 

(8) The quotient of two numbers equals their difference. 

(9) A certain number increased by 1 equals another number 
diminished by 3. 

4. Translate into words these symbolic expressions. For 
example, (1) means "one-ninth of the difference between C times 
a certain number and its square": 



(1)^; 


(«)%-' = %"'; 


(11) y/mn=^ >/m \/n\ 


(2)^-;S 


(7)%i = ^t!; 
^ y y + 1 


(12)^-^^ = 1. 
y/x— y 


(3)^^ 


(8)^ = ».»; 


(13) a^V = (ai)«; 


(^)^; 


(») F -^ 


(1*)^ + 1 = (^)'+1' 


(^) .^2^ 


<-)? = (?)'' 


(15) y/a + l^y/a + 1. 



5. Find the number whicli may be put in place of the letter 
in each of these equations to furnish true equations : 

(l)2a: + 3 = 7; 

Solution.— 

2a? + 3 = 7 
3 = 3 

2x = 4 by Prindtple II, 

2 2 

a; = 2 by Principle IV. 
Check'. 207 + 3 = 2x2 + 3 = 4 + 3 = 7, which is correct. 
(2)3.r-l= 5; (4) ^7:+l = 2; («) ^+f = V; 
(3) 8a;-G = 18; (5) ^f -2 = 1; (7) -^ -1= V- 
Note.— First multiply both sides of (7) by 21. 

6. i'ind the value of the numbers of problem 1 (8), (9), and 
(10). 



3oG RATIOXAL GRAMMAR SCHOOL ARITHMETIC 

§202. Problems. — for either arithmetic or algebra. 

1. The mercury column in a thermometer rose a certain num- 
ber of degrees one day, and 3 times as many degrees the next day. 
It rose 12° during the 2 days. How many degrees did it rise 
each day? 

Arithmetical Solution.— 

A certain number denotes the rise the first day. 

3 times this number denotes the rise the second day. 
Hence 4 times a certain number denotes the rise in two days. 

4 times a certain number equals 12° (by the given problem). 
Once the number equals 3°, the rise the first day (Principle IV). 

3 times the number equals 9°, the rise the second day (Principle III). 
CJieck . 3° + ^° = 12°, the rise in two days. . 

Algebraic Solution.— 

Let X denote the first day's rise. 
Then, 3x denotes the second day's rise. 
X + 3x denotes the rise in 2 days. 
4x = 12°. 

X = 3°, the first day's rise (Principle IV). 
3x = 9^, the second day's rise (Principle III). 
Clieck: 3° + 9° = 12°. 

2. A man bought 4 times as many hogs as cows, and after 
selling 5 hogs he had 23 hogs left. How many cows did he buy? 

Arithmetical Solution.—- 

A certain number represents the number of cows bought. 

4 times this number represents the number of hogs bought. 

4 times this number minus 5 denotes the number of hogs left. 

Then 4 times this number, minus 5, equals 23 (by the problem). 

4 times this number equals 23 plus 5 (Principle I). 

4 times this number = 28. 

This number = 7, the number of cows (Principle IV). 
Check: 4 X 7 — 5 = 23. 
Algebraic Solution.— 

Let X denote the number of cows bought. ! 

Then, 4x denotes the number of hogs bought. 

4x — 5 denotes the number of hogs left. 

Then 4x — 5 = 23 (by the problem). 

4x = 28 (Principle I). 

X = 7 (Principle IV). Ans. 7 cows. 

Check: 4 X 7 — 5 = 23. 

3. Two masses were placed on one scale pan of a balance and 
found to weigh 18 lb. One of the masses was then placed in each 
pan, and it required 4 lb. additional on the light pan to balance 
the scales. What was the weight of each mass? 



USE OF LETTEBQ TO BBPBE8ENT NUMBERS 357 

Arithmetical Solution.— 

A certain number of pounds denotes the weight of the heavier mass. 

Another number of pounds denotes the^weight of the lighter mass. 

The first number plus the second number denotes the combined 
weight (18 pounds). 

The first number minus the second denotes the difference of the 
weights, or the additional weight, which equals 4 pounds. 

2 times the first number equals 18 plus 4 equals 22. 

The first number equals 11. 

11 plus the second number = 18. (Principle IV). 

The second number = 7. (Principle II). 

The weights are, then, 7 lb. and 11 lb. 

Check: 11 lb. + 7 lb. = 18 lb. 
11 lb. — 7 lb. = 4 lb. 

Algebraic Solution.— 
Let X denote the number of pounds in the weight of the heavier mass. 
Let y denote the number of pounds in the weight of the lighter mass. 
Then, x-^-y denotes the number of pounds in the combined weight, =1S 
X — y denotes the number of pounds in the additional weight, = 4. 

x + y = lS 
X — y = 4 

2x = 22 (Principle I). 
X =11 (Principle IV). 
11 + y = 18 
2/ = 18 — 11 = 7 (Principle II). 

Check: 11 + 7 = 18 
11 — 7= 4 

Solve the following problems by both the algebraic and tho 
arithmetical method, and state which of the solutions is the 
shorter: 

4. A man cut for me 3 times as many ash trees as oak trees, 
and as many hickory trees as ash trees and oak trees together. 
In all he cut 32 trees. How many of each kind did he cut? 

5. A man bought 4 times as many 2^ stamps as 5^ stamps, 
and i as many 10^ stamps as 2^ stamps. For all he paid 14.95. 
How many of each kind did he buy? 

Note. — Let x denote the number of 5^ stamps purchased. 

6. The side of a rectangular field is twice its breadth and the 
distance around the field is 240 rd. How many acres does the 
field contain? 

7. I bought a number of books one day, 4 times as many the 
next day, and 3 books the third day. In all I bought 33 books. 
How many did I buy each day? 



358 BATIONAL GRAMMAR SCHOOL ARITHMETIC 

8. A newsboy sold a certain number of papers on Wednesday, 
twice as many on Thursday, and 3 more on Friday than on 
Thursday. In all he sold 48 papers. How many did he sell 
each day? 

Note.— Let x denote the number of papers sold on Wednesday. 

9. Texas lacks 17,470 sq. mi. of being large enough to make 
5 states the size of Illinois. How large is Illinois if Texas con- 
tains 265,780 square miles? 

10. Maude has 78^, which lacks 18jJ of being 6 times as much 
money as James has. How many cents has James? 

11. It is twice as far from Chicago to Niles by a certain route 
as from Niles to Albion, and from Chicago to Albion is 190 mi. 
How far is it from Chicago to Niles? 

12. Rochester, N. Y., is 17 mi. more than twice as far from 
Chicago as is Detroit. Eochester is 587 mi. from Chicago. How 
far is Detroit from Chicago? 

13. Niagara Falls is 8 mi. less than 9 times as far from Chicago, 
as is Michigan City. Niagara Falls is 514 mi. from Chicago. 
How far is it from Chicago to Michigan City? 

14. It is 4 times as far, less 16 mi., from Chicago to N. Y. 
City as from Chicago to Ann Arbor. If it is 976 mi. from Chicago 
to N. Y. City, how far is it to Ann Arbor? 

15. Champaign is 5 mi. more than ^ as far from Chicago as 
is Cairo. If it is 128 mi. from Chicago to Champaign, how 
far is it from Chicago to Cairo? 

16. A father and his son together earn $75 a month, and the 
father earns 4 times as much as the son. How much does each 
earn? ^ 

17. William has 3 times as many marbles as Joseph, and both 
together have 24. How many marbles has each? 

18. A line 32 in. long is to be divided into 2 parts, one of 
which is 8 in. longer than the other. How long is each part? 

19. A tree 75 ft. high was broken by the wind so that the part 
standing was 15 ft. longer than the part broken off. How long 
was each part? 

20. The area of a rectangular field is 6400 sq. rd. and its 
length is 160 rd. ; what is its width? 



USE OF LETTERS TO REPRESENT NUMBERS 



359 



21. The area of a rectangular sidewalk is 120 sq. rd. and the 
width is 1^ yd. ; how long is the walk? 

22. A man received $180 for 72 days' work; what was his 
daily wage? 

23. Find the side of a square field equal in area to a rectangle 
80 rd. long by 20 rd. wide. 

24. A man received $3000 for 80 A. of land ; how much did 
he receive per acre? 

25. The circumference of a circle is 2200 rd. ; what is the 
radius? (Equation: 6f r = 2200.) 

26. The area of a circle is 201^^ m.*; how long is the radius? 
the circumference? 

27. Find the radius and the circumference of a circle whose 
area is 1 square inch. 

§203. Formal Work. 

Find the value of the unknown quantity, x or y, in each of the 
following : 

1. 6a; + 4 = 22. Note.— First multiply both mem- 

2. 3^-10=14. bars of 12 by a.. ' 

3. 4y-2.v + 2 = 16. 

4. 8^4-^ + 3 = 21. 

5. |a;-2 = 16. 

6. |a;+5 = 35. 

Note. — First multiply both mem- 
bers by 3. 

8.^^ = 2. 



9. -g— - 5. 

10. 6(a; + 2)=72. 

11. !^=13. 

12.^-15. 

X 



13. 


25 
2a; 


= 5. 






14. 


x-V 


3a; +12 
11 


= 4. 




15. 


1x- 


-2a; + a; 
5 


= 18. 




16. 


21- 


-a; =33 


-2a;. 




Note.— First add 2a5 to both ; 
bers, then subtract 31 from 
members. 


mem- 
both 



17. 12a; +4= a; + 90. 

4 

18. 7rc-^ = 2a; + 39. 

o 

19. .1^ . 3. 

20. ?^^ = 7. 



3G0 RATIONAL GRAMMAR SCHoOL ARITHMETIC 

21. In the following equations W denotes the weight of the 
rope, or cable, in lb. per yd. ; Z the working load in tons; /S, the 
hreakinij load in tons, and 2), the diameter of the rope, or cable, 
in inches. 

For hemp cable, W^.b11D^\ Z = .109Z)^ >S'= .654Z>«. 

For tarred hemp cable, W^ 1.0362^^; L = .247i>2; S= 1.480i>^ 

For manila rope, W^ .7652^^ L = .329/)^; 8= 1.8772>^ 

For iron wire rope, W^ 3.847Z>'; L = 2.8622^^; S=11.0\2D\ 

For steel wire rope, W= 3.9462^^; L = 4.4412^^; 8^ 27.6302>^ 

Find the weight per yd., the working load and the breaking 
load of a hemp cable of 1" diameter; of 2^" diameter. 

22. Solve similar problems for tarred hemp rope ; for manila 
rope. 

23. Find TT, 2^, and 8 for an iron wire rope for which D = 1\"\ 

24. For the same values of 2>, find PT, Z, and 8 for steel wire 
rope. 



§204. Equations Containing Two Unknown Numbers. 

In these problems two numbers will be unknown. The facts 
of each problem will furnish two equations. From these facts 
write both equations. Then change one or both equations, or 
combine them, in accordance with Principles I-V, pp. 350-2. 

PROBLEMS 

1. The sum of two numbers is 10 and their difference is 4. 

Find the numbers. 

Solution. — Let x stand for the larger number. 
Let y stand for tlie smaller number. 
Then 

(1) x + y=10 (why?) 
and 

(2) x-y= 4 ('vhy?) 

Add Equations (1) and (2) member to member and write- 

(3) 2x =14. By which Principle? 
Then 

(4) X = 7. 

Which Principle enables us to get Equation (4) from Equation (3)? 
Putting this value, 7, of x, in place of x in Equation (1), we have 

(5) 7 + 2/ = 10 

Subtracting 7 from both sides of Equation (5), we get 

(6) y= 3 



EQUATIONS CONTAINIl^G TWO UNKNOWN NUMBERS 361 

By which Principle do we have the right to subtract 7 from both 
sides of Equation (5) and then write the two differences equals 

The two desired numbers are then 7 and 3. 

N.B. — The j)roblem shows how to find any two numbers when their 
sum and their difference are known. 

2. ThjB sum of two numbers is 18 and their difiference is 6. 
Find the numbers. 

3. A piece of wire 8 ft. long is cut into two pieces, one of which 
is 2i ft. longer than the other. How long is each piece? 

4. A 16-oz. mixture of acid and water contains 9 oz. more 
water than acid. What is the weight of each part of the mixture? 

5. In a school of 40 children there are 12 more girls than 
boys. How many girls are in the school? How many boys? 

6. The side of a rectangular lot is 75 ft. longer than the end 
and it is 350 ft. around the lot. Find the length, width and area 
of the lot. 

7. The sum of two weights is 40 lb. and their difference is 8 
lb. Find the weights. 

8. I am thinking of two numbers. Their sum is 23 and their 
difference is 8. What are the numbers? 

9. The top end of a May-pole 20 ft. high broke so that the part 
left standing was 5 ft. longer than the part broken off. How 
long was each piece? 

10. A mixture of 18 oz. of vinegar and water contained twice 
as many ounces of vinegar as water. How many ounces of each 
liquid was in the mixture? 

Solution. — Equation (1) x + y = 18. Explain how this equation is 

obtained. 
Equation (2) x = 2y. How obtained? 

Write 2y in place of x in Equation (1), and have dy = 18. Find y. 
From Equation (2) then find x. 

Put the numbers found for x and y in Equation (1) in place of x and y 
and notice whether the result is atrue equation. This is called a' 'check. ' * 

11. A mixture of water and acid weighing 28 oz. contains 
3 times as much water as acid. How many ounces of each does 
the mixture contain? 

12. A silver dollar weighs 412.5 grains and it contains 9 times 
as much pure silver as alloy. Find the weight of pure silver and 
of alloy in a silver dollar. 



362 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

13 A piece of bronze weighing 15 lb. is made of copper and 
tin. It contains 4 times as much copper as tin. How mUch of 
each metal is there in the piece? 

14. A bronze gun, weighing 25 tons, is made of copper and 
tin in the proportion of 9 times as much copper as tin. How 
much of each metal is there in the gun? 

15. How many cubic inches of pure acid and of water are 
needed to make a gallon (231 cu. in.) of acid solution 12^% pure, 
i.e. containing 12^% of pure acid and 87^% of water? 

16. The sum of one number and twice another is 11. The 
difference of the two numbers is 5. Find the numbers. 

17. The sum of two numbers is 13. Twice the first number 
minus 3 times the second is 1. Find the numbers. 

18. The sum of a number and half another number is 8^. 
Twice the first number minus 3 times the second is 5. Find the 
numbers. 

19. Divide 24 in. into two parts, one of which is 6 in. greater 
than the other. 

20. Divide 32 into two parts, one of which is 3 times the other. 

21. Divide 28 into two parts, one of which is J of the other. 

22. Two numbers are to each other as 3 to 5 and their sum 
is 64. Find the numbers. 

23. If 1 be added to the numerator of a certain fraction it 
becomes i; but if 1 be added to the denominator, the fraction 
becomes J. Find the fraction. 

24. Three times one number added to 2 times another number 
equals 23. Also 2 times the first number minus the second num- 
ber equals 6. Find the numbers. 

Solution. — ^The equations are Sx-\-2y = 23 
and 2a; — 2/ = 6 

Multiply both sides of the second equation by 2 and get 4x — 2y = 13. 
Now add this to the first equation and have 7x = 35. 
From this find x, then substitute the value of x in the second equation 
and find y. 

Check with the first equation. 

25. Eight times a number minus another number equals 4. 
Three times the first number plus 5 times the second equals 66. 
Find the numbers. 



USES OF THE EQUATION 362 

Suggestion. — Multiply the first equation through by 6 and add the 
second equation. 

26. Two numbers are to each other as 5 to 7 and their sum 
is 6. Find the numbers. 

27. Two numbers are as 2 to 7 and their difference is 25. 
Find the numbers. 

28. One man said to another, "Give me one of your sheep and 
I'll then have as many as you." The other replied, "No; you 
give me one of yours and I'll have twice as many as you." How 
many sheep had each? 

Find the numbers the letters stand for in the following and 
check your work. 



r""^ . 33.| 



Sx - 2y = 20 

30.i^ + r'« 34.j«^ + ^ = « 

(a:- 2^ = 8 \x -y = 3 

\x + y = 15 I 6a + 26 = 17 

_ i2x + dy=^d0 ^^ ( 3c + 6? = 21 

^^•|2:.-y = 10 ^^1^-2^ = 2 



USES OF THE EaUATION 

§205. Thermometers. 

Fig. 250 shows the way the thermometer tube is marked off and 
numbered on the three thermometers most used by civilized 
countries. The Fahrenheit, or common, scale is on the left; 
the Centigrade scale, used everywhere in scientific work, is on 
the right; and the Reaumur (Ro'mur) scale, much used in Ger- 
many, is between the other two. 

On all scales the fundamental points are the freezing-jwint 
where the top of the mercury column stands when the bulb is 
in water containing melting ice, and the boiling-point, where the 
top of the mercury column stands when the bulb is in water 
commencing to boil. For any temperature the reading is the 
number which belongs to the mark on the scale at the top of the 
mercury column. 



364 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



REAUMUR 

rlOO AgphaUum nuUa 




Figure 350 



1. What is the reading for 
"Water boils" on the Fahren- 
heit scale? on the Centigrade 
scale? on the Reaumur scale? 

2. What are the three scale 
readings for "Water freezes"? 

3. Into how many equal 
parts is the space between the 
boiling-point and the freezing- 
point divided on the Reaumur 
scale? on the Centigrade? on 
the Fahrenheit? 

4. The lengths of those small 
spaces are marked off on the 
tubes above the boiling-point 
and below the freezing-point 
and numbered, as shown in 
Fig. 250. Notice how the 
numbers run on each of the 
scales between the freezing and 
the boiling points; below the 
freezing point. How should 
the numbering continue above 
the boiling point? 

5. What do we call a change 
of temperature sufficient to 
expand, or contract, the mer- 
cury column an amount equal 
to one of the short spaces on 
the Fahrenheit scale? on the 
Centigrade scale? on the Reau- 
mur scale? 

These are written 1°P., 
1°C., and 1°R., and are read 
"1 degree Fahrenheit," "1 
degree Centigrade," and "1 
degree Reaumur." 



USES OF THE EQUATION 



365 



C. If the readings above (f are written +1°, +2°, +3°, +5°, etCc , 
how should we write the readings below 0°? 

7. What do the + and — then tell us about the readings? 

8. Point out these readings in the figure: +2° F. ; +36°C.; 
+29°E.; -14°C.; -20°R.; -38°F.; -10°F. 

9. Give the readings on each scale for the mean temperature 
at the Equator; at Rome; London; Chicago; Edinburgh; St. 
Petersburg; at the Poles. 

10. Which degree is the longest, the 1°P., the 1°C., or the 
1°R.? Which is the shortest? 

11. How many Fahrenheit degrees are equal to 80 °R.? How 
many Centigrade degrees equal 180°F.? How many Reaumur 
degrees equal 100°C.? 

12. On a sunny day find how many Fahrenheit degrees warmer 
it is in the sun than in the shade? Give the C and R readings 
corresponding to the two F readings and also the O and R differ- 
ences. 

§206. Applied Algebra (Graduation of Thermometers). 

The problems of §202, p. 356, show how the 
use of the equation simplifies some kinds of arith- 
metic problems. The work given here will illus- 
trate how some problems that would be quite, 
difficult by arithmetic can be easily handled by 
means of the equation. 

Experiments. — To convert thermometer 
readings from one scale to another: 

Three thermometer tubes, exactly alike, were 
filled with mercury to the same height. Their 
bulbs were immersed in a vessel containing melt- 
ing ice. Points were marked on the tubes at the 
tops of the mercury columns; beside the point 
on the first tube was written 32, and beside those 
on the second and third tubes was written. 

All three of the bulbs were then immersed in 
water commencing to boil, and the tops of the 
mercury columns were marked; beside the mark 
on the first 212 was written; beside that on the 
second, 100; and beside the mark on the third, 
80. Let AB of Fig. 251. denote the straight line passing through 
the tops of the three mercury columns when the bulbs are in 



r 


If 


J j 


1 


<L 


F12-- 


ioo*- 3 


X_ 




Y 


A_ 






_B 




32' 


^ 


0' 




1 


1 1 


1 



F C R 

FIOI7BE 251 



3G6 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

melting ice. Let CD denote the line passing through the tops 
of these columns when the bulbs are in boiling water. 

Suppose the space lying between the line AB and the line CD 
on the first tube to be divided into 180 equal spaces and call each 
of these small spaces one degree Fahrenheit (written 1°F.). 
Denote its length (in in. or in cm.) by/. 

Suppose this same space on the second tube between the lines 
AB and CD to be divided into 100 equal spaces, and call one of 
these smaller spaces one degree Centigrade (written 1°C.). 
Denote its length (in in. or in cm.) by c. 

Divide the space on the third tube between the lines AB and 
CD into 80 equal spaces, and call one of these spaces one degree 
Reaumur (written 1°R.). Denote the length of this space 
(in in. or in cm.) by r. 

Notice carefully that no two of the lengths /, c, and r are 
equal. Denote the length of the space from AB io CD by 8. 

1. 8 equals how many times/? c? r? 

2. Since 180/ equals 8^ and 100 c equals 8^ how must 180/ 
compare with 100 c? (Answer this by writing an equation.) 

This problem exemplifies a new principle of much importance 
in using equations: 

Principle VI. — Numbers that are equal to the same number 
are equal to each other. 

3. Since 180/ equals Sy and 80 r also equals 8, how must 180/ 
compare with 80 r? 

4. If 180/ = 100c, /equals how many times c? 

5. If 180/ = 80 r, / equals how many times r? 

6. From these two equations, giving the value of /, write 
another equation by the aid of Principle VI. 

7. Explain the meaning of the following equations: 

(l)/=|c; (2)/=|r, and(3)c = |r. 

Now suppose spaces equal to / marked off on the first tube 
above 212 and below 32, spaces equal to c marked off on the second 
tube above 100 and below 0, and spaces equal to r marked off on 
the third tube above 80 and below 0. The first tube is then grad- 
uated to the Fahrenheit scale, the second, to the Centigrade soale, 
and the third, to the Reaumur scale. 



USES OF THE EQUATION 3G7. 

§207. Equivalent Beadings on the Three Thermometers. 

On all thermometers, when the tops of the mercury columns 
are above 0, the readings are called positive^ and are marked +; 
when below zero they are called negative, and are marked -. 

If the thermometers are exposed to the same temperature 
between that of melting ice and of boiling water, the tops of the 
mercury columns will stand in a straight line, as XY. Let the 
mark that stands by this line at the top of the column on the first 
be called F; on the second top, C; and on the third, R. 

1. Denoting by S the distance from the line AB to the line 
XF show by Fig. 251 that (1) 8=f (i^-32); (2) >S'= Cfc; and 
{3)8= Br. 

2. Show by Principle VI that (4) / {F- 32)= cC; (5) / {F- 32) 
= rE; and (6) cC=rR, 

3. Show from equations (1), (2), and (3), §206, problem 7, 
that (7) c= I/; (8) /' = |/ and (9) r= |6-, by using Principles III 
and IV. 

4. In equation (4) problem 2 substitute c = |/ from (7) problem 
3 and by Principles IV and II show that (I) F=^ ^Ch-32. 

5. In a similar way show that the following equations are true ; 

(II) ^=172 + 32; (V) R = i {F-32); 

(III) ry= 1(^-32); {Vl)Ji = iC. 
{lY)C=iR; 

Note. — F, C, and R may be regarded as standing respectively for any 
• Fal ' • ~ • ' 



corresponding readings on the Fahrenheit, Centigrade, and Reaumur 
thermometers. 

§208. Problems. 

1. Convert the following into their F. and R. equivalents by 
the aid of the proper equations (I) to (VI). 

MELTING TEMPERATURES ON CENTIGRADE SCALE. 

Ice 0.0° Zinc 412.0° 

Benzol + 4.4° Antimony 432.0" 

Tallow 43.0° Silver 1000.0° 

Paraffin 46.0° Copper 1100.0° 

Wax 62.0° Gold 1200.0° 

Sulphur 115.0° Cast iron 1200.0° 

Tin 230.0° Cast steel 1375.0° 

Bismuth 250.0° Wrought iron 1600.0° 

Cadmium 320.0° ?l»tinura 1775.0° 

Lead 826.0^ Iridium 1960.0* 



368 



RATIONAL GRAMMAR SCHOOL ARITHMBTIO 



2. Convert these Fahrenheit boiling temperatures into their 
G. and R. equivalents : 



Ether +95° 

Carbon Bisulphide +114.8° 
Sulphuric Acid ... +14° 
Chloroform +141.8° 



Alcohol +172.4° 

Water +212° 

Mercury +674.6° 

Zinc + 1904° 



3. Convert any observed Fahrenheit readings to their Centi- 
grade and Reaumur equivalent readings. 

4. The heat of the body of a healthy man is 37.2°C. What 
are the Fahrenheit and Reaumur equivalents? 

5. Solid carbonic acid dissolves in ether and the solution is a 
liquid of -90°C. What is the Fahrenheit equivalent? 

Solution.— In equation (I) if C = -90^ |C = -162". Then 162° meas- 
ured downward and 32° measured upward, or, -162^ *+ 32^= -130°. 

6. For every vapor and gas there is a so-called critical temper- 
ature above which it remains in the gaseous condition, no matter 
how high the pressure upon it. The following are the critical 
temperatures of some vapors and gases: Ether vapor, +196°C. ; 
carbonic acid, +31° C; etheline, +9°C.; oxygen, — 118°C. ; nitro- 
gen, -145°C., and hydrogen, -174°C. What are the Fahrenheit 
equivalents? 

Laws of Thermometer Bepresented Graphically. 

When it is necessary to change 

a great many readings from one 

scale to another it is convenient to 

plot the equations. To illustrate 

this, take the equation ^= f C+- 32. 

Draw two perpendicular lines, as 

OX and OY, Fig. 252. Let all 

Centigrade readings be measured off 

horizontally and the corresponding 

Fahrenheit readings vertically to 

-^ any convenient scales. 

— The equation shows that if 

50 CENTIGRADE RLA0INC5 C= 0°, F= 32° ; if C= 20°, F^ 68°; 

and so on. In this way we fill out 

FiGUBB 253 the following table : 




8H0BTENIKG AKD CHECKING CALCULATIONS 3C9 



0. 


F. 


C. 


F. 


0° 


32° 


100° 


212° 


20° 


68° 


200° 


392° 


40° 


104° . 


-20° 


-4° 


50° 


122° 


-40° 


-40° 


75° 


167° 


-80° 


-112° 



Note. — ^The numbers preceded by the sign — are readings below 
zero, and such numbers for C. must be measured off from toward the 
left, and for F. they must be measured off downward. 

Study the points on Fig. 253 and note whether they seem to lie on a 
straight line. With a ruler draw a straight line through these points. 

For minus (— ) values of C, as for C= —20, proceed thus: F=lx 
(—20) + 32 = 9 X (— ^ff°)-h 32 = 9 X (— 4) + 32 = —36 + 32. But 36° meas- 
ured downward and then 82° measured upward is the same , as 4° 
measured downward, or F= —4°, if 0= — 20"; and similarly for other 
minus ( — ) values of C. 

1. Measure off the value = 60°, then measure vertically 
upward to the line drawn through the points, thus obtaining the 
corresponding Fahrenheit reading. What is F. for 0. = 60°? 

2. Similarly, find from the drawing the values of F. corre- 
sponding to these values of C: 120°; 160°; 180°; 220°; -60°; 
-100°; 110°. 

3. Make a similar table and plot of one or more of the other 
five equations of problem 5, p. 367. 



METHODS OF SHOBTENING AND CHECKING 
CALCULATIONS 
§210. Illustrations. 

1. Additions, subtractions, multiplications, and divisions are 
conveniently checked by casting out the 9's, as explained on pp. 
28, 54, and 75. 

2. To check against gross errors (blunders), first think through 
the problem, making rough mental calculations with numbers 
that are approximately correct, and decide about what the result 
must be, 

3. Check by performing reverse operations; that is, check 
addition by adding columns in reverse order ; check subtraction 
by adding the subtrahend to the remainder; check division oi 
square and cube roots by multiplication, and so forth. 



370 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

The second rule may be illustrated by a few problems: 

1. Find the value of 15f A. of land at $87^. 

Think thus: 16 A. @ $90 would be worth $1440; at $85, 16 A. would be 
worth $80 less, or $1360. At $87i, 15^ A. of land is worth J of |85 less 
than 11440 — J of $80 (=$1400), or about $1383. The exact computation 
gives $1382.50. 

Or thus: 87i = I of 100. Therefore 15 J X 87i = 1580 x 5 = 7 X 197.5 
= $1382.5. Aws. $1382.5. 

2. On June 13, 1903, wheat is quoted in Chicago at 76f^ per 
bushel. Find the cost of 150 bu. at this price. 

3. 250 shares of Illinois Central R. R. stock sold at $135^ a 
share. For how much did they sell? 

4. The diameter of a circular rod is If". How many inches 
in the circumference pf a right section of the rod? How many 
square inches are there in the area of a right section of the rod? 
(For an approximation use w = 3i^.) 

5. The outside diameter of a circular hollow iron tube is 2^" 
and the inside diameter is 2". How many cubic inches are there 
in a 12' length of the tube? 

6. A distance was measured with a chain 98.75 ft. long and 
was found to contain 38.75 lengths of the chain. The chain was 
supposed to be 100' long. How great an error was made in 
measuring the line by using the supposed length? 

§211. Shortening and Checking Addition. 

1. The noon temperatures on 7 successive days were 66**, 54**, 
44°, 62°, 66°, 79°, 88°. Find the average for the week. 

66 Mental Work. 70 + 50 + 44 = 164; 164 + 60 = 

54 224; 224+2 = 226; 226 + 70 = 296; 296 — 4 = 292; 

44 164 292 + 80 = 372; 372 — 1 = 371; 371+90 = 461; 461 — 

226 62 2 = 459. 

66 292 "^1^^^ ^^ called two-column addition. A little prac- 

391 79 tice will make this method useful for 1, 2, or 3 columns 

g^ of figures. It may be used to advantage to check addi- 

— — tions'made in the ordinary way. 

72459 Another method in use by expert accountants is to 

65 6** Ans group the figures into sums of 10, 20, 30, and so on. 

Thus, in the given problem, 8 + 2. 6 + 4, 6 + 4, and 9 

make 39. Then 7 + 3, 8 + 6 + 6, 6 + 4. 5, are 45. Sum, 459. 

2. Write a few two and three column addition problems and 
practice these methods until you can use them rapidly. 



SHORTENING AND CHECKING CALCULATIONS ^Hl 

§212. Makingup Method of Subtraction. 

1. A paying- teller in a bank had $5485 in his cash drawer in 
the morning, and during the day he paid out the following 
amounts: $37.50; $1G5.75; $10.25; $3.50; $2.88; $1.76; 
$05.17; $908.23; $3.67. How much money remained in the 
drawer? 

CONVENIENT FORM. MENTAL WORK.— Add the first column, 

Total, $5485.00 thinkiog thus: 10, 23, 31, 41. 41 and 9 make 

— zrzz the next larger number than 41 ending in 

^^I'^l (the first figure in the total) Write the 9 in 

^Jxlf *^^® result and add the 5 into the second col- 

^^'l^ umn. Then 11, 21, 34, 48. 48 + 2 = 50, the 

^•22 next number larger than 48 which ends in 

*-^2 (the second figure of the total). Write 2 in 

_1J2 ^1^® result and add 5 into the third column. 

J2nl Then, 21. 31, 39. 39 + 6 = 45. Write 6 and 

^^•23 add 4 into fourth column. Then 10, 17, 26. 26 + 

^'^" 2 = 28. Write 2 and add 2 to next column. 



$4226.29, balance Then, 12. 12 + 2 = 14, and finally, 1 + 4 = 5. 
Write the 4. 
The advantage of this method is that it foots all the numbers and 
subtracts their sum, at once, as the numbers stand in the account book, 
from the total, giving the balance directly. 

2. A bank customer's deposit at the beginning of the month 
was $398.75. During the month he drew out the following 
amounts: $16.75^ $1.75; $5.25; $12.87; $1J!8.32; $40.45; 
$2.18; $9.16; $1.57; $11.38; $12.62. Find the customer's 
balance at the end of the month. 



§213. Shortened Multiplication. 
1. Multiply 73 by 67. 

g^ = 7Q i 3 j 73 X 67 = (70 + 3) (70 -3) = 702 + ;|J x 3 - ? X ;r|J - 3^ = 
702 _ 32 = 4900 — 9 = 4891. 

Algebraic form : (a + &) (a — 6) = a" — 61 

This applies to finding the product of any two numbers the sum of 
whose units is 10 and whose tens digits differ by 1. 

Rule. — Find the difference bettueen the square of the tens and 
the square of the units in the larger number. 



372 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

2. Find these products by the rule: 

(1) 34 X 26. (4) 58 x 42. (7) 71 x 69. 

(2) 22 X 18. (5) 65 x 55. (8) 99 x 81. 

(3) 43 x 37. (6) 98 x 82. (9) 95 x 85. 

3. Find the square of 38. 

38« = (30 + 8)2 = 302 + 2 X 8 X 30 + 82 = 1444 
Algebraic form: (a + 6)2 = a2 + 2a6 + b*. 

Show the correctness of the following rule : 

Rule. — Square the tens, double the product of the tens hy the 
units and square the units, then add the three results. The sum 
is the square of the number. 

4. Square these numbers by the rule : 

(1) 3G. (3) 64. (5) 19. (7) 125. 

(2) 47. (4) 58. (0) 95. (8) 148. 

Note.— Call the 12 in (7) 12 tens; also cjall the 14 in (8) 14 tens. 

36 might be written 40 — 4. Then 362 = (40 — 4)2, (40 — 4)2 = 402 - 
2 X 4 X 40 + 42. A Igebraic form : (a—hf = a'^ — %ab + 62. Make a rule 
for squaring 36 in this form. 

When long decimals are to be multiplied and the product is 
required to only a few decimal places, contracted multiplication is 
of great advantage. The next problem illustrates this sort of 
multiplication. 

5. The radius of a circle is 238. 3G ft. What is the length of , 
the circumference to the second decimal place, or to the nearest 1 
.01 foot? 

Note.— -The length of the diameter is 476.72 feet. 



Common Form 

476.72 
3.1416 


Shortened Form 

476.72 

6141.3 digits reversed 


28 

47 

1906 

4767 

143016 


6032 
672 

88 
2 


1430.16 

47.67 

19.07 

48 

29 



1497.66 



3552 1497.67 



SHORTENING AND CHECKING CALCULATIONS 373 

Explanation. — ^In the shortened form the digits of the multiplier are 
written in reverse order, and the units digit is always written under that 
decimal place in the multiplicand which is to be the last one retained in 
the product. 

Multiply by units digit first, then by tens, and so on; in each case 
begin the multiplication by an^ digit with the digit just above it in 
the multiplicand. Begin the writing of each partial product in the same 
vertical line on the right. 

Note. — It is necessary on beginning to multiply by any digit ,to 
glance at the product by the preceding digit of the multiplicand to see 
how many units are to be added into the product by the digit just above. 
Thus, the multiplication by 4 would begin with 6, but 4 times the pr«* 
ceding digit (7) is 28, and this being nearly 8, the product 4x6 would 
be increased by 3, giving 27. 

Expert computers use the shortened form altogether. 

6. Find the following products to the second decimal place by 
the method of shortened multiplication: 

(1) 36.428x3.1416. (3) 7.8843x1.0863. 

(2) 186.086 X 108.336. (4) 168.7431 x 28.329. 

7. Find these products to .001 by shortened multiplication 

(1) 36.1872x6.8734. (3) 629.3865x3.1416. 

(2) 128.63x3.8629. (4) 1284.683x3.1416. 

§214. Shortened Division. 

1. Divide 648.7863 by 68.372 to the nearest .01. 
Convenient Form 

9.49— Quotient Explanation.— Find the units digit of 

68.37)648.7863 the quotient in the usual way. Then cut off 

_^15_?2__ one digit from the right of the divisor and 

33 45 find the next digit of the quotient, then cut 

27 35 off another digit from the divisor, etc. A dot 

— — - is sometimes placed over each digit in the 

6 10 divisor as it is set aside. 

6 15 

2. Find the following quotients to two decimal places: 

(1) 1786.786 -*. 3.1416. 

(2) 632.068 + 8.6249. 

(3) 1206.3862 + 28.3762. 

(4) 865.28476 4- 361.2946, 



374 



ItATlOKAL GRAMMAR SCHOOL ARITHMETIC 



§215. Shortened Square Eoot: Square Boot by Subtraction. 



261 



4 



46 
f 521 



284 
276 



832 
521 



522)311.9(586 

2670 

449 

417 



32 



Square root = 261.586 



1. Find the square root of 68432.93628 
correct to the third decimal figure. 

Explanation.— By placing a dot over every 
alternate digit in the given number, counting 
both ways from the decimal point, we obtain 
as many dots as there are to be figures in the 
square root. Find half the figures by the ordinary 
method of square root, and the rest by contracted 
division as shown. The result will very nearly 
equal the square root. 

2. Find, by the shortened method, to two 
decimals the square roots of the following: 

(1) 6432. 18G4. (2) 38629.72468. (3) 
8764.932651. 



3 subtractions 



Square root may be found by subtraction, after noticing that 

1 =1^ l + 3 = 2«; 1 + 3 + 5 = 3% 1 +3 + 5 + 7 = 4«; and so on. 

It is seen that the square root of the sum of the odd numbers 

in order from 1 upward is equal to the number of odd numbers 

added. An example will illustrate the use of this principle: 

3. Extract the square root of 104976. 

. . . Explanation. — Place a dot over each 

104976 ( 324=square root alternate digit beginning on the right, 
thus separating the number into groups 
of 2 digits each. From the first group on 
the left subtract 1, then 3, then 5, and so 
on as shown, until the remainder is less 
than the next odd number. The number 
of subtractions is the first root digit. In 
the present case it is 3. 

Bring down the next two-digit group. 
Double the root digit foimd and annex 
1 to it, and subtract, then replace the 1 by 
2 subtractions ^ *^d subtract, etc. The number of sub- 
tractions indicates the next root digit. 

Study the remaining steps and learn 
how to proceed further. This method 
gives the exact square root and may be 
used to check results found in the ordinary 
way. Some computing machines are based 
647 on this method of obtaining the square 

647 4 subtractions root. 

4. Find the square roots, by subtraction, of the following: 

(1) 1156. (3) 174.24. (5) 54,756. 

(2) 44,944. (4) 49,284. (6) 289,444. 





1 


9 
3 
6 
5 


61 


149 
61 




88 
63 


641 


2576 
641 




1935 
643 


1292 
645 



SYNOPSIS OF DEFINITIONS 

PAGE 

6. The average of two or more numbers is their sum divided by the number 

of them. (C/. pp. 85, 132.) 
8. A board foot is a board one foot long, one foot wide, and not more than 
one inch thick. 
10. Cash rent of farming land is a stated amount of cash per acre. 
Grain rent of fanning land is a stated part of all the crop. 
The tenant farmer is one who raises his crop on another man's fann. 
12. Normal here means average. (Cf. p. 13.) 

The mean of two numbers is half their sum. 
17. The digits (or figures) are the ten characters, 0, 1,2,3,4, 5,6,7,8,9. 
The name value of a digit is the value depending only upon the name of 

the digit. 
The place value of a digit is the value depending only upon the place 
of the digit. 
22. Addition is combining numbers into a single number. 
The sum or amount is the result of the addition. 
The addends are the numbers to be combined or added. 
26. The displacement is the number of tons of water a floating vessel 

pushes aside. 
33. Subtraction means either of two things: 

(1) The way of finding the difference, or remainder, of two 
numbers. 

(2) The way of finding either one of two addends when their sum 
and the other addend are known. 

With the first meaning, the number from which we subtract is 
the minuend. The number to be subtracted is the subtrahend. The 
result is called the difference or remainder. 

With the second meaning, the known sum is the minuend. The 
known addend is the subtrahend. The unknown addend is the 
difference or remainder. 

43. Literal numbers are those that are denoted by letters. 

44. Multiplication of whole numbers is a short way of finding the sum of 

equal addends when the number of addends and one of them are 
given . 

The given addend is the multiplicand. 

The number of equal addends is the multiplier. 

The result or sum is the product. 

When a problem is expressed in an equation, the equation is 
called the statement of the problem. 

375 



376 BATIOXAL GRAMMAR SCHOOL ARITHMETIC 

The number which may stand in place of x in an equation is 
called the valv^ of x. 
A factor of a given whole number is one of two or more whole numbers, 
which, multiplied together, produce the given number. (C/. pp. 51, 
142.) 
52. A number which has factors other than itself and 1 is called a com- 
posite number. (C/. p. 143.) 
A niunber which has no factors other than itself and 1 is a prime 
number. (C/. p. 143.) 
57. One inch of rainfall means one cubic inch of water for each square 
inch of horizontal exposed surface. 
The number of cubic units (cu. in., cu. ft., cu. yd., etc.) a vessel holds, 
when full, is called its capacity. 

61. Division of whole numbers is a short way of subtracting one number 

from another a certain number of times in succession. (C/. p. 62.) 

62. Division is a way of finding one of two numbers when their product 

and the other number are given. 

The product is called the dividend. 

The given number is the divisor. 

The required number is the quotient. 

We may say also that the dividend is the number to be divided, 
the divisor is the number by which the dividend is measured or 
divided, and the quotient is the measure. (C/. p. 61.) 
78. A one-brick wall is a wall one brick thick, the bricks lying on the largest 
surfaces, the sides being exposed. 

85. To average means to find the average. 

86. The range of temperature is the difference between the highest and the 

lowest temperatures. 

97. Such an expression as y = 10 is called an equation. 

The number on the left of the sign of equality is called the first mem- 
ber, or the left side, of the equation. The number on the right is 
called the second member, or the right side, of the equation. 

98. The pencil point of the compasses is called the pencil-foot or the 

pen-foot. The other point is called the pin-foot. 
A circle is a curve such as is drawn with compasses. 
An arc of a circle is a part of a circle. 
The center of a circle is the point where the pin-foot of the compasses 

was placed to draw the circle. 
A diameter of a circle is a straight line joining two points of the circle 

and passing through the center. (C/. p. 107.) 
A radium of a circle is a straight line joining its center and a point of 

the curve. 
Circles whose centers are at the same point are called concentric 

circles. 



SYNOPSIS OF DEFINITIONS ' 377 

101. Dividing a line into two equal parts is called bisecting the line. 

102. An equilateral triangle is an equal-sided triangle. 

An isosceles triangle has at least two sides equal. (C/. p. 103.) 
That side of an isosceles triangle not equal to either of the two equal 
sides is called the base of the isosceles triangle. 

103. A scalene triangle has no two sides equal. 

104. A regular hexagon is a regular six-sided figure. 

105. The sum of all the bounding lines of a figure is called the perimeter of 

the figure. 
Money is the common measure of the value of all articles that are 
bought and sold. 
107. One of a whole number of equal parts of a given magnitude is called 
a fractional unit. (Cf. p. 138.) 

109. A representation of an object, showing the various parts as folded 

back and spread out on a flat surface, is called a develop- 
ment. 

110. A parallelogram is a quadrilateral whose opposite sides are parallel. 

(Cf. pp. 282, 283.) 
A rectangle is a parallelogram whose angles are (equal) right angles. 

(Cf. pp. 282, 283.) 
A square is both a rectangle and a rhombus. 

111. The altitude of a parallelogram is the distance square across. 

The altitude of a triangle is the shortest distance to the base from the 
opposite comer. With all except isosceles triangles any side may 
be regarded as the base. 

118. The mean solar day is the average time interval during which the 

rotation of the earth carries the meridian of a place eastward from 
the sun back around to the sun again. It is the average length of 
the interval from noon to the next noon. (Cf. p. 217.) 

119. A section of land is a tract one mile square and containing 640 acres. 

120. A township is a tract of land six miles square. 

124. Per cent means hundredth or hundredths. 

125. Interest is money paid for the use of money. (Cf. p. 252.) 
132. The average rate of running is the distance divided by the time. 

135. The ratio of one number (or quantity) to a second number (or quantity) 

is the quotient of the first number (or quantity) divided by the 
second. 

136. The ratio of one magnitude to a second magnitude is called also the 

measure of the first by the second. 
The result of measuring one number by another is called the numerical 

measure of the first by the second. 
An equation of ratios is called a proportion. 
138. The fractional unit of a fraction is one of the equal parts expressed by 

the fraction. (Cf. p. 107.) 



378 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

139. The number above the fraction line is called the numerator (meaning 
numberer). 

The number below the fraction line is called the denominator (mean- 
ing namer). 

The numerator and the denominator are together called the terms of 
a fraction. 

141. A fraction is said to be in its lowest terms when the numerator and 

the denominator are the smallest possible whole numbers without 
changing the value of the fraction 
The greatest common divisor (G. C. D.) of two numbers is their greatest 
exact common divisor. 

142. Two numbers that have no common factor, except 1, are said to be 

prime to each other. 
A factor of a number is an exact divisor of the number, or a divisor 
that is contained without a remainder. (Cf. p. 44.) 

143. A number that has no factors except itself and 1 is called a prime 

number. (Cf. p. 52.) 
A number that has factors besides itself and 1 is called a composite 

number. (C/. p. 52.) 
Any number that can be exactly divided by the number 2 is called 

an even number. All other whole numbers are called odd numbers. 

147. A denominator which is common to two or more fractions is called a 

V common denominator, (Cf. p. 140.) 
When a common denominator is the least number that can be found 
which may be used as a common denominator of the fractions, it is 
called the least common denominator (L. C. D.). 

148. A number that can be exactly divided by another nuitiber is called a 

multiple of the latter number. 
A number that can be exactly divided by two or more numbers is 
called a common multiple of those numbers. 

149. The least common multiple (L. C. M.) of two or more numbers is the 

least whole number that is exactly divisible by each of the numbers. 

151. A proper fraction is a fraction whose numerator is less than its denom- 
inator. 
An improper fraction is a fraction whose numerator is equal to, or 

greater than, its denominator. 
A mixed number is a number that is composed of an integer and a 
fraction. 

160. To multiply a whole number by a fraction means to divide the multi- 
plicand into as many equal parts as there are units in the denom- 
inator, and to take as many of these equal parts as there are units 
in the numerator of the multiplier. 

170. Dividing 1 by any number, whole or fractional, is called inverting the 
number. 



SYNOPSIS OF DEFINITIONS 379 

The reciprocal of any number is the number inverted. 
172. Fractions containing fractions in one or both terms are called complex 
fractions. 

The outside terms of a complex fraction (such as -y) are called the 

extremes, and the inside terms are called the means. 
176. A straight line connecting the mid-point of a side of a triangle with 
the opposite comer is called a median of the triangle. 
The vertex of an angle is the comer. (The plural of vertex is vertices 
(v^r'-tX-ses.)) (C/. pp. 274, 288.) 

178. To trisect a magnitude is to divide ijt into three equal parts. 

179. A right-angled triangle is called a right triangle. 

180. Parallel lines are lines running in the same direction. 

182. A square is inscribed in a circle if the vertices of the square are all on 

the curve. 
186. The first, second, third, and fourth numbers of a proportion are called 
the first, second, third, and fourth terms of the proportion. 
The first and the fourth terms of a proportion are called the extremes, 

and the second and the third terms, the means. 
The first two terms of a proportion are together called the first couplet; 
and the third and the fourth terms, the second couplet. 

190. A dot, called the decimal point, or point, is used to show the units* 

digit. The point always stands (or is supposed to stand) just to the 
right of the units' digit or place. 

191. The unit of the 1st place, or digit, to the right of the decimal point is 

called the tenth; of the 2d place, or digit, the hundredth; of the 3d, 
the thousandth; of the 4th, the ten-thousandth; and so on. 

192. A decimal fraction, or decimal, is a fraction whose denominator is 

10, 100, 1000, or some power of ten, in which the denominator is 
not written but is indicated by the position of the decimal p>oint. 
A power of 10 here means a number obtained by using 10 as a factor 
any whole number of times. 

193. A pure decimal is a decimal whose value is less than one. 

A mixed decimal is a decimal whose value is greater than one. 
Numbers expressed in both decimals and common fractions are called 

complex decimals. 
A simple decimal is expressed without the use of common fractions. 
Finding the sum of decimal numbers is called addition of decimals. 
195. Finding the difference of decimal numbers is called subtraction of 

decimals. 
198. By the number of decimal places of a number is meant the number of 

digits (zero included) on the right of the decimal point. 
204. The distance round a circle is sometimes called the circumference of 
the circle. (Cf. p. 107.) 



380 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

207. The specific gravity of any solid or liquid substance is the ratio of its 

weight to the weight of an equal bulk of water. 
209. Decimals that do not tenninate are called non-terminating decimals. 
Non-terminating decimals that repeat a digit or group of digits 
indefinitely are called repetends, or circulating decimals, or circulates. 
211. A right section is a section made by cutting squarely across. 
213. A denominate number is a number whose unit is concrete. 
A concrete unit is a unit having a specific name. 

A compound denominate number is a number expressed in two or 
more units of one kind. 
215. A perch of stone is a square-cornered mass, l'Xli'Xl6i' = 24i cubic 
feet. 
A cord of firewood is a straight pile, 4' X 4' X 8'= 128 cubic feet. 
A cord foot is a straight pile of wood, 4'X4'Xl'= 16 cubic feet. 
217. A leap year is a year of 366 days. 

A common year is a year of 365 days. 
225. Reduction from higher to lower denominations is called reduction 
descending. 
Reduction from lower to higher denominations is called reduction 
ascending. 
229. A m£ter is approximately one ten -millionth of the length of the part 
of a meridian of the earth, that reaches from the equator to the 
pole, called a quadrant of the earth's meridian. 
231. An are is one square dckameter. 
A liter is one cubic decimeter. 

A gram is the weight of one cubic centimeter of distilled water at the 
temperature of its greatest density (39.1° Fahrenheit). 
236. The number written before the sign "%" is called the rate per cent. 
The number, together with the sign "%," is called the rate. (Cf. 
p. 252.) 
239. The result of finding a given per cent of any amount, or number, is 
called the percentage. 
The amount, or number, on which the percentage is computed is 
called the base. 
243. Elevation means height above mean sea level. 

248. Commission is a sum of money paid by a person or firm, called the 

principal, to an agent for the transaction of business. It is usually 
reckoned as some per cent of the amount of money received or 
expended for the principal. 

249. A shipment of goods sent to an agent to be sold is called a consignment. 
A commercial (or a trade) discount is a certain rate per cent of re- 
duction from the listed prices of articles. The discount is usually 
allowed for cash payments or for payment within a specified time. 
(Cf. p. 260.) 



SYNOPSIS OF DEFINITIONS 381 

252. Interest is money charged for the use of money. It is reckoned at a 
certain rate per cent of the sum borrowed for each year it is bor- 
rowed. (C/. p. 125.) 

When money earns 3, 6, 7, or 10 cents on the dollar annually (each 
year) the raie is said to be 3%, 6%, 7%, or 10% per annum (by the 
year), and the rate 'per cent is said to be 3, 6, 7, or 10. (C/. p. 236.) 

The sum of money on which interest is computed is called the principal. 

The principal plus the interest is called the amount. 

259. A promissory note is a written promise, made by one person or party, 

called the maker ^ to pay another person or party, called the payee, 

a specified sum of money at a stated time. 
The sum of money for which the note is drawn is called the face value, 

or the face, of the note. 
The date on which the note falls due is called the date of maturity, and 

the time to run from any given date is the time yet to elapse before 

the note falls due. 

260. Discount is a deduction from the amount due on a note at the date 

of maturity. (C/. p. 249.) 
The sum of money which, at the specified rate and in the time the note 

is to run before falling due, will, with interest, amount to the value 

of the note when due, is called the present worth of the note. 
The difference between the value of the note, when due, and the 

present worth is called the true discount. 
The bank discount of a note is the interest upon the value of the note 

when due, from the date of discount until the date of maturity. 

261. When a note or bond is paid in part the fact is acknowledged by the 

holder by his writing the date of payment, the sum paid, and his 
signature on the back of the note or bond. This is called an in- 
dorsement. 
264. The small wheels under the front of a locomotive engine are called 
engine truck wheels, or leaders. The large wheels are called drivers. 
The smaller wheels just behind the drivers are called trailers. 

266. Tractive force is pulling (or drawing) force. 

267. Replacing a letter (in an equation) by a number is called substituting 

the number for the letter. 
Performing the operations indicated in an equation and obtaining 

the number-value of a letter is called finding the value of that letter. 

274. The surfaces of the cube meet each other in edges, thus forming lines. 

The corners are called vertices. A single comer is a vertex. (Cf. pp. 

176, 288.) 
The edges meet each other in comers of the cube, thus forming points. 
278. When two lines meet, making the angles at their point of meeting 

(intersection) equal, the lines are said to be perpendicular to each 

other, and each is called a perpendicular to the other. 



382 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

The angles thus formed are called right angles. 
279. Lines which go through the same point are called concurrent lines. 
282, 283. A quadrilateral is a four-sided figure. 

282, 283. A rhombus is a parallelogram whose sides are equal. 

283. A rhomboid is a parallelogram that is not a rectangle. 

A trapezoid is a quadrilateral having at least one pair cf opposite 

sides parallel. 
285 A quadrant of arc is one quarter of a complete circle. 

An angle is the amount of turning of a line about a point as a pivot. 

It may also be regarded as the difference of direction of two lines. 
A straight angle is the sum of two right angles. 

286. An angular degree is one of the ninety equal parts of a right angle. 
A degree of arc is one of the ninety equal parts of a quadrant. 

A sextant is one-sixth of a complete circle. 
An octant is one-eighth of a complete circle. 

287. A perigon is an angle equal to one complete revolution. 

288. If two lines intersect each other, two angles lying opposite to each 

other are called opposite or vertical angles. 
The lines which include an angle are called the sides of the angle 
The point where the sides meet is called the vertex of the angle. (C/. 

pp. 176, 274.) 

289. An angle that is smaller than a right angle is called an acute angle. 
An angle that is larger than a right angle and less than a straight 

angle is called an obtuse angle. 
291. Two angles whose sum equals a right angle, or 90®, are called com- 
plemental angles. Two angles whose sum equals two right angles, 
or 180°, are called supplemental angles. 

297. An imaginary curved line, called the equaior^ divides the surface of 

the earth into the northern and the southern hemispheres. 

298. The imaginary curved lines running east and west are called parallels; 

those runnmg north and south are called meridians. {Cf. pp. 
299, 310). 

299. Longitude is the distance in degrees, minutes, and seconds (of arc) 

measured on a parallel due eastward or westward from a chosen 
meridian called the prime meridian. {Cf. pp. 297, 300.) 
Latitude is a similar distance measured northward or southward on 
a meridian. 

305. The date line is the 180th meridian. 

306. A four-sided figure (a quadrilateral) having at least one pair of parallel 

sides is called a trapezoid. 
The altitude of a trapezoid is the distance square across between two 
parallel sides called the bases of the trapezoid. 

307. The lengths of the bases and of the altitude of a trapezoid are called 

its dimensions. 



SYNOPSIS OF DEFINITIONS 383 

A square of roofing is a ten -foot square (100 sq. ft.). 

Shingles are said to be laid so many inches to the weather vrhen the 

lower end of each course of shingles extends so many inches below 

the course next above it. 

310. A toiVnship is a tract of land six miles square. 

A range is a tier (or row) of townships running north and south. 

311. A section is a tract of land one mile square. 

Such of the north and the west rows of half-sections of a township as 
do not have exactly 320 acres each are called lots. 

312. The volume of any figure is the number of cubical units within its 

bounding surfaces. * 

317. A straight line connecting two points of an arc is called a chord of 

the arc. 

318. A circle is said to be circumscribed around a triangle and the triangle 

is said to be inscribed in the circle if all the vertices of the triangle 
are on the curve. 

319. The longest side of a right triangle, that is, the side opposite the right 

angle, is called the hypothenuse, 

321. ^The product obtained by using any number twice as a factor is called 

the square of that number. 

322. The square root of a number is one of its two equal factors 

326. The cube of a number is the product obtained by using the niunber 

three times as a factor. 

327. The cube root of a number is one of its three equal factors. 

328. In triangles having the same shape, angles lying opposite pro- 

p>ortional sides in different triangles are called corresponding angles. 
In triangles having the same shape, sides lying opposite equal angles 

are called corresponding sides. 
336. The amount paid for insurance is called the premium. 

The written agreement between an insurance company and the 

insured is called a policy. 
The amount for which the property is insured is < aVed the Jace of 

the policy. 
338. A tax is a sum of money levied by the proper officers to defray the 

expenses of national, state, county, and city governments, and for 

public schools and public improvements. 
Assessed valuation means the estimated value of the property that is 

assessed. 
341. A stock (called also a stock certificate) is a written agreement made by 

a company to pay the holder a certain part of the earnings of the 

company. 
When a stock company pays to the holders of its stock $2 on each 

$100 of its capital stock, or 2% on its stock, the company is said 

to be paying a %2 dividend, or a 2% dividend. 



384 RATIONAL GRAMMAR SCHOOL ARITHMETIC 

At par means at its face value. 

Above par means at more than its face value. 

Below par means at less than its face valu^. 

A bond is a written agreement made by a national, state, county, or 
city government, or by a company, to pay the holder interest at 
a stated rate on a stated sum of money, called the face of the bond. 

A broker is a man that makes a business of buying and selling stocks 
and bonds for other people. 

A broker's charge for his services is called brokerage. 

342. Preferred stocks are stocks which pay a fixed dividend before any 

dividends are paid on common stocks. 
Dividends are paid upon common stocks after expenses and dividends 

on preferred stock have been paid 
Government 2s are government bonds paying 2% interest. 

343. Compound interest is interest on interest. 

The compound amount is the principal plus the compound interest. 
345. Coupon notes are notes, attached to interest-bearing notes, for the 
amount of interest due each at interest-paying period. 

349. An expression in :which the sign < or > stands between two numbers 

is called an expression of inequality. 

350. The signs =, < , and > are called relation signs. 

General Defilnitions 

Quantity is limited magnitude. 

Number is the result of the measurement of quantity. 

Number may also be defined as the ratio of quantities of the same kind. 

Arithmetic is the science of numbers and the art of using them. 



GENERAL INDEX 



Explanatory. — The following index has been prepared for daily use by 
young students of^^various ages, and the principal purpose underlying its 
construction has been to enable all studfents to find readily and quickly 
the thing desired, that it may be in use not occasionally but continually. 
To this end the entries have been arranged not merely for the noun (or the 
principal noun) but also, in many cases, for other words of the phrase^ thus 
making the index usable also by students to whom grammatical distinctions 
are not entirely familiar. 

In reading, the phrase after the comma (where one occurs) is often to 
be read first; thus "addition, short methods for" is read "short methods 
for addition." 

The references are to pages. Where two page references are sepc,rated 
by a dash the meaning is "and included pages;" thus 27-30 means pages 
27, 28, 29, 30. 

Angle and of arc, minute of, 287, 289 
Angle and of arc, second of, 287, 289 
Angle, sides of, 288 
Angle, vertex of, 176, 288 
Angles, addition of, 289-293 
Angles and arcs, measuring, 285-289 
Angles (and figure), opposite or verti- 
cal, 288 
Angles, complemental, 291 
Angles, difference and sum of, 289-293 
Angles, division of, 176 
Angles formed by parallel lines and a 

line cutting them. 288 
Angles of hexagon, sum of, 292 
Angles of n-gon, sum of, 293 
Angles of octagon, sum of, 293 
Angles of parallelogram, sum of, 292 
Angles of quadrilateral, sum of, 292 
Angles of right triangle, sum of acute, 

292 
Angles of similar triangles, correspond- 
ing, 328 
Angles of triangle, sum of. 291 
Angles, subtraction of, 290 
Angles, sum and difference of, 289-293 
iUigles, supplemental, 291 
Angular degree, 286, 289 
Angus cattle, weights and prices of, 203 
Annually, 252 
Apothecaries' weight, 214 
Applications of percentage, 336-345 
Applications of proportion, practical, 

188-190, 327-336 
Applications to transportation problems, 

see also tractive force, 264-271 
Arabic notation, 20 
Arabic numeral, 20, 21 
Arc, 98, 285 



Above par, 341 

Account, forms of, 5, 47-49, 88-91 

Accounts, bills and, 88-96 

Accounts, farm, 47-50 

Accounts rendered, 91 

Acquired territory of United States, 

area and dates of, 246 
Acre, 215 
Acres covered by one mile of furrow of 

given width, number of, 161 
Acute angle (and figure), 289 
Addend, 22 

Addition, checking, 24, 28, 369, 370 
Addition of angles, 289-293 
Addition (of common numbers), 21-32 
Addition of decimals, see also decimals, 

193-195 
Addition of fractions, see fractions 
Addition of fractions having common 

fractional unit, 146 
Addlfion, short methods for, 369, 370 
Admission of states, territories, etc., 

dates of, 39 
Air, pressure oL 127, 128 
Almanac, 244, 245 

Altitude, see also elevation and height 
Altitude of parallelogram. 111 
Altitude of trapezoid, 306 
Altitude of triangle. 111 
Amount, 22, 252 
Amount, compound, 343 
*'And" In decimals, use of, 191 
Angles, 285 

Angle (and figure), acute. 289 
Angle (and figure), obtuse, 289 
Angle (and figure), right, 278, 289 
Angle (and figure), straight, 285 
An^Ie and of arc, degree of, 286, 289 



885 



386 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Arc, degree of angle and of, 286, 289 
Arc, minute of angle and of, 287, 289 
Arc, second of angle and of, 287, 289 
Arcs, measuring angles and, 285-289 
Are (unit of land measure), 231 
Area, 31, see also measuring surface 

and mensuration 
Area and population of States and Ter- 
ritories, 30, 35 
Areas of common forms, 132-134 
Areas of countries, 35, 09. 70 
Areas of fields, 9, 10 
Areas of river basins, 37 
Ascending, reduction, 225 
Assessed valuation. 338 
Average, 6, 85. 132 
Avoirdupois weight, 214, 220 



Bale {unit of paper measure), 217 

Bale of cotton, weiglit of, 161 

Ball, velocity of cannon, 83, 206 

Ball, velocity of rifle, 206 

Bank discount, 260 

Barrel, number of gallons in, 34, 228 

Barrel, number of pecks In, 71 

Base (in percentage), 239 

Base line, 310 

Base of isosceles triangle, 102 

Base of parallelogram. 111 

Baseball diamond (and figure), 2, 3 

Bases of trapezoid, 306 

Basins, areas of river, 37, 247 

Battleships, data of, 26 

Beans, capacity for absorbing water, 

195, 196 
Beef, food parts of, 190 
Below par. 341 
Billion, 18, and footnote 
Bills and accounts, 88-96 
Birds, speeds of, 75, 206 
Bisect, 101, 176 

Bisector, figure of perpendicular, 101 
Bisector, perpendicular, 176 
Block and lots (and figure), town, 76 
Blocks, figure of city streets and, 306 
Board foot, 8, 227 

Board measure, see measuring wood, 8 
Body, average surface of human. 128 
Body, dailv requirement of water and 

of solid food for human, 145 
Body, normal temperature of a man*s, 

368 
Boiling point. 116, 363 
Boiling temperatures, 364, 368 
Bond, face of, 341 
Bond quotations and transactions, stock 

and. 341 
Bonds, stocks and. 341-343 
Book, figure of, 184 
Box, capacity of square-cornered, 57, 

58, 112 
Breeze, kinds of. 14 
Brick wall, kinds of, 78, footnote 
Brick work. 78-80, 308, 309 
Broker. 341 
Brokerage. 341 
Bulk, see volume 

Bundle (unit of paper measure), 217 
Bnshel, 210 



Bushel, number of cubic feet of grain 

In, 112, 205 
Bushel, number of cubic inches in, 113 
Bushel of various articles, weights of, 

71, 114, 216 
Bushel, Winchester, 216 
Butcher's price list, 4 

Cables, data concerning, 360 
Calculations, methods of shortening and 
checking. 51-53, 60, 68, 70-75, 191. 
192, 369-374 
Calendar, Gregorian, 217 
Calendar, Julian, 217 
Calendar month, 217 
Calendar year, 217 
Cancellation, 72, 82, 83. 158, and note 
Cannon ball, velocity of, 83, 200 
Capacity. 57 

Capacity, measuring, 112, 113, 210. 231 
Carat, 220 

Carrier pigeon, speed of, 206 
Cars, engines and tenders, heights, 
lengths and weights of railroad. 15, 
64, 129. 130, 264 
Cash rent, 10 
Casting out the nines, 28, 53, 54, 75, 

369 
Cattle at Chicago Stock Yards, receipts 

of, 35, 36 
Cattle, weights and prices of Angus, 203 
Cattle weights and values of, 50 
Cent, 213 
Centare, 231 
Center of circle, 98 
Center of square, 320 
Centeslmi, 214 
Centi-, 2.30 

Centigrade thermometer, 303-309 
Centime, 214 
Central time, 303 
Centre. See center 
Century, 217 
Certificate, stock. 341 
Chain, also square chain, 215 
Chair, figure of. 184 
Change of notation, 21 
Checking calculations, methods of short- 
ening and, 369-374 
Checking calculations : ' 

Addition, 24. 28, 369, 370 

Casting out the nines, 28, 53, 54, 
75, 369 

Division. 68. 74. 75. 369. 373 

Multlnlication. 53, 54, 309, 371-373 

Square root, 374 

Subtraction. .S69, 371 
Cheat expansion. 106 
Chest measure. 12 
Chicago Stock Yards, receipts of cattle 

at. 35. .SO 
Chord (of a circle), 317 
Circle (and figure). See also ctrcum- 

ference-on<t mensuration. 98, 289 
Circle and parallelogram compared (and 

figures). 210 
Circle, area of, 210, 211 
Circle, center of, 98 
Circle, hour, 804 ^ 



GENERAL INDEX 



387 



Circles, concentric, 98 
Circulate, 209 
Circulating decimal, 209 
Circumference. See also circle and men- 
suration, 107, 204, 205, 289 
Circumference to diameter, ratio of, 204, 

205 
Circumferences of various tilings. See 

also diameters, etc., 56, 106-108, 

159, 2D4, 205, 211 
Circumscribed, 318 
Cities, populations of, 28, 41 
City streets and blocks, figure of, 306 
Clock, figure of, 285 
Coal, hard vs. soft, 4 
Coal in ton, number of cubic feet of 

hay and of, 205 
Coast lines of continents, lengths of, 

246 
Coffee Imports of United States, 27, 56, 

203 
Coins, fineness of United States gold 

and silver, 55, 226, 361 
Coins, weights of United States, 50, 55, 

155, 156, 194, 195, 226 
Commerce. 38-41 
Commission, 248, 249 
Common denominator, 140, 147 
Common fraction. See fractions 
Common fraction, reduction of decimal 

to, 192, 193 
Common fraction to decimal, reduction 

of. 208, 209 
Common multiple, 148 
Common stock, 342 
Common uses of numbers, 127-134 
Common year, 217 
Comparison of prisms, 276-278 
Compasses (and figure), pen-foot and 

pin-foot of, 98 
Complemental angles, 291 
Complex decimal, 193 
Complex fractions, 172, 173 
Complbsite factors, 142, 143 
Composite number, 52, 143 
Compound amount, 343 
Compound denominate numbers, 213-235 
Compound interest, 343-345 
Concentric circles, 98 
Concrete unit, 213 
Concurrent lines, 279-281 
Cone, model and development of right 

circular, 314 
Cone, volume of right circular, 314 
Consignment, 249 

Constructions. {See also model and de- 
velopment) : 

Drawings: 

I. Circle with given radius, 98 

II. Line equal to given line, 98, 99 

III. Line equal to sum of two or 
more given lines, 99 • 

IV. Line equal to difference of two 
given lines, 100 

V. Line equal to two. three or four 
times given line, 100 

VI. and I. Divide given line Into 
two equal parts (bisect It), 100, 
101, 176 



VII. Equilateral triangle with each 
side equal to given line, 102 

VIII. Isosceles triangle with sides 
equal to two given lines, 102 

IX. Scalene triangle with sides 
equal to three given lines, 103 

X. Three-lobed figure in circle of 
given radius, 103, 104 

XI. Regular hexagon in circle of 
given radius, 104 

I. Divide given line Into two equal 
parts (bisect it), 176 

II. Divide given angle into two 
equal parts (bisect it), 176 

I. Line parallel to given line, 177 

II. Line parallel to given line and 
through given point, 178 

III. Line parallel to given line and 
through given point, 178 

IV. Divide given line Into three 
equal parts (trisect it), 178 

V. 30** and 60 ^ and 45% right 
triangles, 180 

VI. Through given point, line paral- 
lel to given line, 181 

VII. Through given point on given 
line, perpendicular to line. 182 

VIII. Through given point out of 
given line, perpendicular to line, 
182 

I. Perpendicular to given line from 
given point on It, 272 

II. Perpendicular to given line 
from given point out of it, 272, 
273 

III. Square in circle of given radius, 
273 

Greasings (paper-folding) : 

I. Perpendicular to given line, 
through given point on It, 278 

II. Bisect an angle, 279 

III. Three non-parallel lines, 279 

IV. Bisect three angles of triangle, 
279 

V. Bisect given line. 280 

VI. Square and Its diagonals, 280 

VII. Three perpendiculars from 
vertices of triangle to opposite 
sides, 281 

VIII. Three perpendicular bisectors 
of sides of triangle, 281 

IX. Rectangle and Its diagonals, 
281 

Cuttinos: 

Find sum of two given angles. 290 

Find difference of two given angles, 
290 

Find sum of angles of equilateral 
triangle, 291 

Find sum of angles of scalene tri- 
angle, 291 

Find sum of angles of right trian- 
gle, 292 

Find sum of acute angles of right 
triangle, 292 

Find sum of angles of quadrilateral. 

Find sum of angles of hexagon. 292 
Find sum of angles of octagon. 293 
Find sum of angles of n-gon, 293 



388 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Find relations of parts of rectangle 
with a diagonal, 293 

Find relations of parts of parallel- 
ogram with a diagonal, 2d3 
Dratoings: 

I. Center of given arc, 317, 318 

II. Bisect given arc, 318 

III. Circumscribe circle about equi- 
lateral triangle, 318 

IV. Trefoil, 318 

V. Quatrefoil, 318 

VI. Designs of figure 216, p. 319 

VII. Sixfoil, 319 

VIII. Right triangle, 319 

IX. Right triangle having given 
hypothenuse, 319 

X. Right triangle having given two 
legs, 320 

Cuttings: 

XI. Find relation of squares on 
sides of isosceles right triangle, 
320 

XII. Find relation of squares on 
sides of any right triangle, 320. 

Constructive geometry. See also con- 
structions, 98-105, 176-182, 272- 
281, 289-293, 307-320 

Copeck, 214 

Cord {unit of fire-wood measure), 215 

Cord foot. 215 

Corresponding angles of similar trian- 
gles. 328 

Corresponding sides of similar triangles, 
328 

Cost of living, 3-6, 93-96 

Cotton, weight of bale, 161 

Counting, 217 

Counting, measurement by, 217 

Countries, areas and populations of, 70 

Couplets, first and second, 186 

Coupon notes, 345 

Creasing paper. See a2«o . constructions, 
278-281 

Critical temperatures of gases, 368 

Crow, speed of, 75, 206 

Crown, 213, 214 

Cuba, area of, 35 

Cuban school data. 26 

Cube, model and development, of three- 
inch, 274 

Cube of number. 326 

Cube root of number. 327 

Cube roots, cubes and, 326, 327 

Cubes and cube roots, 326, 327 

Cubes of units and tens, table of. 326 

Cubic feet of hay and coal In ton, num- 
ber of, 205 

Cubic foot, number of gallons In, 266 

Cubic foot of various things, weights 
of. 56, 64. 82, 114, 161, 165, 207, 
208, 235, 266 

Cubic foot of water, weight of, 165. 235, 
266 

Cubic Inch of steel, weight of. 113 

Cubic Inch of water, weight of, 164 

Curves, plotted, 86, 120. 122. 267, 368 

Cylinder, lateral surface of right circu- 
lar, 313 

Cylinder, model and development of 
right circular, 313 



Cylinder of engine, figrnre of, 211 
Cylinder, volume of ri£^t circaUr, SU 

Dairying, 16, 50, 75, 204. See aUo miik 
ana mlikings 

Date line, 305 

Date of maturity, 259 

Dates of acquisition of territory o! 
United States, 246 

Dates of admission and population of 
states, territories, etc., 39 

Day, 21/ 

Day, mean solar, 118, 217 

Decl-, 230 

Decimal, 192 

Decimal, circulating. 209 

Decimal, complex, 193 

Decimal fraction. See decimal, 192 

Decimal, mixed, 193 

Decimal, non-terminating. 209 

Decimal notation, 19, 190. 191 

Decimal places, number of, 198 

Decimal point, 190 

Decimal point in product, 198, 199 

Decimal, pure. 193 

Decimal, reduction of common fraction 
to, 208, 209 

Decimal, simple, 193 

Decimal to common fraction, reductloa 
of, 192, 193 

Decimals, 190-212 : 
Notation, 190, 191 
Numeration, 191, 192 
Reduction of decimal to common frac- 
tion, 192, 193 
Addition, 193-195 
Subtraction, 195-197 
Multiplication ( pointing off the prod- 
uct), 198. 199 
Division, 200-208 

Of decimal by integer, 200-202 
By decimal, 200-208 
Reduction of common fraction to deci- 
mal, 208. 209 

Decimals, addition of, 193, 194 

Decimals, division of. See decimals, 
200-203 

Decimals, notation of, 190, 191 

Decimals, numeration of, 191, 192 

Decimals, principles of. See principles, 
etc. 

Decimals, subtraction of 195-197 

Degree, angular, 286, 289 

Degree of angle, 286, 289 

Degree of arc, 286, 289 

Degree of longitude at equator, length 
of, 164 

Deka-. 230 

Denominate number, 213 

Denominate numbers, compound, 213- 
235 

Denominator, 139 

Denominator," common, 140, 147 

Denominator, least common, 147 

Depth, greatest ocean, 69 

Depths of lakes, 248 

Descending, reduction. 225 

Description of land, 310-312 

Development, 109, 275 

Developments. For list see model, et€i. 



GENERAL INDEX 



389 



diagonal, 179/ 292 

:>iagonai divides figure, how, 179, 292, 

293 
Diameter, 98, 107 
Diameter, ratio of circumference to, 204, 

205 
Diameters of eartli, 206, 315, 317 
Diameters of heavenly bodies, 317 
Diameters of various things. See also 

circumferences, etc., 107, 108, 

204, 205, 211 
Diamond (and figure), baseball, 2, 3 
■Difference, 33 

Difference of angles, sum and. 289-293 
Differences of lines, products of sums 

and, 293-295 
Digit or figure, 17 
Digit, name value of, 17 
Digit, place value of, 17 
Dime, 213 

Dimensions of trapezoid, 307 
Discpunt, 249, 260 
Discount, bank, 260 
Discount, trade, 249, 250, 339, 340 
Discount, true, 260 
Discounting notes, 260, 261 
Displacement of vessel, 26 
Distance, measuring, 106-108, 214, 215 
Distances from Chicago to various cities, 

45, 55, 128, 206, 358 
Distances from Washington to various 

cities. 54, 55 
Distribution of population of United 

States, 1900. 29-31 
Dividend, 62, 341 
Divisibility of numbers, 73, 74. See also 

factors, prime and composite, and 

greatest common divisor 
Division and multiplication compared, 

61-63 
Division and subtraction compared, 60, 

61 
Division by multiples of 10, 70-72 
Division, checking, 66, 68, 74, 75, 369, 

373 
Division, long, 65-70 
Division (of common numbers), 60-87 
Division of decimals. See decimals 
Division of fractions. See fractions 
Division of lines and angles, 176 
Division, short, 03. 64 
Division, short methods for, 70-74, 191, 

373 
Divisor, 62 

Divisor, greatest common, 141, 143 
Dollar, 105, 213 
Door, figure of, 183 
Double ea^le, 213 
Dozen, 217 
Dram, 214 

Drivers of locomotive 56, 264, 270 
Dry gallon, liquid and, 216 
Dry measure, 216 
Duck, speed of wild, 206 

Eagle ; half-eagle ; quarter-eagle ; double- 
eagle. 213 

Earth, area of : land area of ; water 
area of, 246 

Earth, diameters of, 206, 315, 317 



Earth, mean diameter of, 315, 317 

Earth, mean radius of, 19, 316 

Earth, velocity of, 83 

Eastern time, 303 

Electricity, velocity of, 206 

Elevation. See also altitude and height 

Elevations and heights of mountains, 

37, 69, 243, 247 
Elevations of lakes, 35, 37, 248 
Elevations of Weather Bureau stations, 

243 
Engine, figure of, cylinder of, 211 
Engines and tenders, heights, lengths 

and weights of railroad cars, 15, 64, 

129, 130, 264, 266 
English notation, 18, and note 
Equal parts, fractions as ratio and as, 

138-140 
Equation, 44, 97 
Equation, first and second members of, 

97 
Equation, left and right sides of, 97 
Equation, principles for using. See also 

principles, etc., 97, 350-352, 366 
Equation, uses of, 348, 360-369 
Equations with two unknown numbers, 

360-363 
Equator, 297 
Equilateral triangle (and figure), 102, 

103, 282, 284, 326 
Equivalent readings of thermometers, 

367-369 
Equivalents, metric and United States, 

230-232 
Equivalents of longitude and time, 305 
Even number, 143 
Expansion, chest, 106 
Expense account, family, 93-96 
Expenses of living, 3-6, 93-96 
Exponent, 143 
Export trade, 1891. 1901, United States, 

40 
Express trains, speed of, 82. 83 
Expression of Inequality, 349 
Extremes, 172, 186, 187 

Pace (of a note), 259 

Face of bond, 341 

Face of policy, 337 

Face value, 259 

Factor, 44, 51, 142 

Factor, prime, 142, 143 

Factors, greatest common divisor by 

prime, 143, 144 
Factors, multiplication by, 51, 52 
Factors, order of, 160 
Factors, prime and composite. See also 

divisibility of numbers, 142, 143 
Factory life upon growth, effect of, 197 
Fahrenheit, thermometer, 116, 363-369 
Falcon, speed of. 206 
Family expense account, 93-96 
Farm accounts, 47-50 
Farm, fencing, 7-9, 133 
Farm, figure of, 7 

Farm products of United States, 83-85 
Farthing, 213 
Fence wire, cost and weight of, 7, 8, 

133, 1,S4 
Fencing farm, 7-9, 133 
Fields, areas of, 9, 10 



390 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Figure or digit, 17 

Filler, 213, 214 

Finding the value of (phrase), 267 

Fineness of United States gold and sil- 
ver coins, 55, 226 

Fire-wood by cord, measuring, 215 

Flat prism, model and development of, 
276 

Folding paper. Bee also constructions, 
278-281 

Food parts of beef, 190 

Food, prices of. 4 

Foot, board, 8, 227 

Foot, cord, 215 

Foot, number of gallons in one cubic, 
266 

Foot; aXao square and cubic foot, 214, 
215 

Foot of compasses, pen-foot and pin- 
foot, 98 

Force. See tractive force 

Forces, Joint effect of, 173-175 

Forms, areas of irregular but common, 
132-134 

Forms, figures of irregular but common, 
132-134, 284 

Forms of account, 5, 47-49, 88-91 

Formula, 239 

Foundation walls, 79 

Fraction as ratio and as equal parts, 
138-140 

Fraction, common, 192. Bee also Frac- 
tions 

Fraction, decimal, 192. Bee aUo deci- 
mals 

Fraction, Improper, 151 

Fraction, lowest terms of, 141-142 

Fraction, proper, 151 

Fraction, reduction of decimal to com- 
mon, 192. 193 

Fraction, terms of, 139 

Fraction to decimal, reduction of com- 
mon, 208, 209 

Fractional numbers, multiplication by, 
54 

Fractional unit. 107, 138 

Fractions, 135-100 

Introduction (ratio and proportion), 
1,35-140 
Ratio (measure), 1.35, 136 
Proportion, i:i«. 137 
Fractions ns ratios and as equal 
parts. 138-140 

Fractions (simple), 140-172 

Reduction to higher, lower and low- 
est terms, 140-142 
Factors, prime and composite, 142, 

143 
Greatest common divisor by prime 

factors, 143. 144 
Addition of frnctions having com- 
mon fractional unit, 146 
Addition of fractions easily reduced 
to common fractional unit, 146, 
147 
Multiples and least common multi- 
ple, 148-151 
Reduction of whole and mixed num- 
bers to fractions, and inversely, 
151-153 



Addition. 153-155 
Subtraction, 155-157 * 
Multiplication, 157-165 

Of fraction by whole namber, 

157-158 
Of mixed number by Whole num- 
ber, 159 
Of whole number by fraction, 

160, 161 
Of fraction by fraction, 162, 163 
Of mixed number by mixed num- 
ber. 163-165 
Division, 165-172 

Of fraction by whole nnmber, 165- 

167 
Of mixed number by whole num- 
ber, 167, 168 
Of any number by fraction, 168- 
172 
Fractions, addition of. Bee also frac- 
tions, 153-155 
Fractions, complex, 172, 173 
Fractions easily reduced to, common 

fractional unit, 146, 147 
Fractions having common fractional 

unit, 146 
Fractions, Introduction to, 135-140 
Fractions, principles of. Bee principles, 

etc. 
Fractions, reduction of. Bee also deci- 
mals and fractions, 140-142, 146, 
147. 151-153, 192, 193, 208, 209 
Fractiotas, subtraction of. Bee fractions 
Fractions to higher, lower and lowest 

terms, reduction of. 140-142 
Fractions with common fractional unit, 

addition and subtraction of, 146 
Franc. 213, 214 
Freezing point, 116, ,363 
Freezing temperatures, 116, 364. 365 
Freight and passenger trains, 128-130 
French notation. 18, and note 
Furnishings, house and, 78-81 
Furrow of given width, number of acres 
covered by one mile of, .161 

G. C. D., 141 

Gain and loss, 240-242 

Gale, 14 

(]rallon, dry gallon and liquid, 216 

Gallon, number of cubic inches in, 113, 
216 

Gallon of milk, number of pounds in 
one, 204 

Gallons In barrel, number of, 34, 228 

Gallons In one cubic foot, number of 
(liquid), 266 

Gases, critical temperatures of, 368 

Gate, figure of, 184 

Geographic mile or knot, 227 

Geography, 37, 38, 68-70, 245-248 

Geometry, constructive. Bee also con- 
structions. 98-105, 176-182, 272- 
281, 289-293, 307-320 

German notation, 18, and note 

Gettysburg address, Lincoln's, 224 

Gill {unit), 216 

(Jlobe, figure* of, 304 

Gold and silver coins, fineness of United 
States, 55, 226 



GENERAL INDEX 



391 



Gold and water compared, weight of, 
165 

Gold, value of one ounce of, 55, 115 

Gold, value of one pound of, 161 

Government 28, etc., 342 

(graduation of thermometers, 365, 366 

Grain (unit of weight) , 214 

Grain rent, 10 

Gram, 231, 232 

Granite, weight of, 56 

Gravity, speclHc, 207, 208 

Great gross, 217 

Great Pyramid, dimensions of, 316 

Greatest common divisor, 141 

Greatest common divisor by prime fac- 
tors, 143, 144 

Gregorian calendar, 217 

Grocer's price list, 4 

Grocery clerk, problems of, 92 

Gross. 217 

Growth, effect of factory life upon, 197 

Growth of people In helpht and in 
weight, 13, 121, 122, 196, 197 

Growth of trees, 147 

Growth of trees, lateral, 147 

Growth of United States, territorial, 
246 

Guinea, 213 

Hawk, speed of, 75, 206 

Hay and of coal in ton, number of cubic 

feet of, 205 
Height, See also altitude and elevation 
Height and weight, growth of people, 

13, 121. 122, 196. 197 ., 
Height to lung capacity, relation of, 12, 

206 
Heights of boys and girls working and 

not working in factories, 197 
Heights of mountains, elevations and, 

37-69, 243, 247 
Heights of persons, 13, 121, 122, 196, 

197, 206 
Heights of railroad cars, engines and 

tenders, 15 
Hekto-, 230 
Heller, 214 
Hexagon (and figure), regular, 104,282, 

326 
Hexagon, sum of angles of. 292 
High land, ratio of low land to, 246 
Higher terms, reduction of fraction to, 

140, 141 
Horse-power, 26, 206 
Horse, speed of, 206 
Hour, 217 

Hour circle of sun, 304 
House and furnishings, 78-81 
House and roof, figure of. 59 
House, figure of end of, 183 
House plans, 78-81 
Houses, walls of, 79 
Hundred-weight, 214 
Hundredth. 191 
Hundredths , measuring by. See also 

percentage and ratio. 123-125 
Hurricane, 14 
Hypothenuse, 319 

lUlnois admitted. 223 



Immigration into United States, 1900, 

1901, 27, 64, 65 
Imports of coffee into United States, 27, 

56. 203 
Imports of molasses into United States, 

27 
Imports of tea into United States, 27, 

56, 203 
Improper fraction, 151 
Inch precipitation, 57, 193 
Inch ; also square and cubic inch, 214, 

215 
Independence Hall, Philadelphia, figure 

of, 274 
Indorsement, 261 
Industrial products, 1897, 1902, values 

of, 40 
Inequality, expression of, 349 
Inscribed square, 182 
Insurance, 336, 337 
Insurance policy, 336 
Integer, 151 

Interest, 125, 126, 252-263 
Interest, compound, 343-345 
Interest, percentage and, 123-126, 235- 

263 
Interest, simple, 125, 126 
Interest table. 257 
Intersection, 99 
Introduction, general, 1-16 
Introduction to fractions, 135-140 
Introduction to ratio and proportion, 

135-137 
Invert, 170 
Isosceles triangle (and figure), 102, 103, 

282, 284 
Isosceles triangle, base of, 102 

Joint efPect of forces, 173-175 
.Julian calendar, 217 
Jupiter, diameter of, 317 

Kilo-, 230 

Kite, figure of, 184 

Knot or geographic mile (nautical unit 
of distance), 227 

L. C. D., 147 

Lakes, areas of, 248 

Lakes, depths of, 248 

Lakes, elevations of, 35, 37, 248 

Land (and figure), section of, 119, 215, 

311 
Land area of the earth, 246 
Land areas of divisions of United States, 

201 
Land, description of, 310-312 
Land, figure of divided section of, 119, 

311 
Land, figure of tract of, 7, 24, 119, 133, 

185, 306, 310, 311, 331-335 
Land, measuring, 119, 120, 133. 134, 

215, 231, 310. 312, 329-336 
Land surveying, 310-312, 329-336 
Lateral growth of trees, 147 
Latitude, 299 

Latitudes of seaports, 299 
Leaders of locomotive, 264 
Leap year, 217 

Least common denominator, 147 ' 
Least common multiple. 149-151 



392 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Leayes of trees, 195, 243 

Legal time. 305 

Length, measuring, 106-108, 214, 215, 
230 

Letters to represent numbers, use of, 
345-369 

Light to travel from moon to earth, 
time for, 55 

Light to travel from sun to earth, time 
for, 55 

Light, velocity of, 19, 51, 55, 206 

Lighted {room), well, 81, 108 

Lincoln's Gettysburg address, 224 

Line, 274 

Line, base, 310 

Line, date, 305 

Line, standard base, 310 

Linear measure, 214, 215, 230 

Lines, concurrent, 279-281 

Lines, division of, 176 

Lines, figure of parallel, 99, 100, 177, 
275, 276, 288 

Lines, parallel, 180 

Link (unit) ; also square link, 215 

Liquid and dry gallon, 216 

Liquid measure, 216 

Lira, 213, 214 

Liter, 231 

Literal numbers, 43 

Literal numbers, subtraction of, 42, 43 

Live stock, a week's receipts and ship- 
ments of. See also cattle, 36 

Living, cost of, 3-6, 93-96 

Living, expenses of, 3-6, 93-96 

Load. Sec tractive force 

Local (sun) time, 300, 305 

Locating places on the earth, 297-299 

Locomotive (and figure). See also en- 
gines and tenders. 264-271 

Locomotive, drivers of, 56, 264, 270 

Locomotive, leaders of, 264 

Locomotive, trailers of, 264 

Locomotives, lengths and weights of, 
129, 130. 264 

Long division, 65-70 

Long ton, 214 

Longitude, 299, 300 

Longitude and time. 300.305 

Longitude and time, equivalents of, 305 

Longitude at the equator, length of one 
degree of, 164 

Longitudes of observatories, 301 

Longitudes of seaports. 299 

liongltudes, table of. 302 

Loss, gain and, 240-242 

Lot (of land) (and figure). 311 

Lots (and figure), town block and, 76 

Low land to high land, ratio of. 246 

Lowest terms of fraction, 141, 142 

Lumber measuring, 8 

Lumber sold by board foot, 8 

Lung capacities, heights and weights of 
men, 206, 207 

Lung capacity, 12 

Lung capacity to height, relation of, 12 

Maker (of a note), 259 
Manufactured products, 1897, 1902, 
values of, 40 



Manufactures, Chicago, 70 

Manufactures, New York City, 1800, 37 

Map of time belts of United States, 303 

Maps of United States, 29, 303 

Map of the world, 296 

Mark, 213, 214 

Marking goods, 251 

Mars, diameter of, 317 

Maturity, date of, 259 

Mean, 12, footnote 

Mean solar day, 118, 217 

Mean solar year, 119, 217 

Mean temperatures. 244, 364 

Means, 172, 186, 187 

Measure, 136 

Measure, chest, 12 

Measure, dry, 216 

Measure, linear, 214, 215, 230 

Measure, liquid, 216 

Measure, numerical, 136 

Measure, surveyors', 215 

Measure, to, 105 

Measurement, 105-126, 136, 138 

Measurements, 31. 32 

Measurements, physical, 11-14. 206, 207 

Measures, metric, 229-232 

Measuring. See also mensuration 

Measuring angles. 285-289 

Measuring arcs. 285-289 

Measuring brick work, 78-80, 308, 309 

Measuring bulk. See measuring volume 

Measuring by counting. 217 

Measuring by hundredths. See also 

percentage and ratio, 123-125 
Measuring by money. See measuring 

value 
Measuring capacity, 112. 113, 216. 231 
Measuring distance. 106-108. 214, 215 
Measuring fire-wood, 215, 232 
Measuring land. 119, 120, 133. 134. 

215, 231, 310, 311, 329-336 
Measuring length, 106-108, 214, 215. 230 
Measuring lumber, 8 
Measuring paper, 217 
Measuring roofing, .3()7-.309 
Measuring stone, 205. 215 
Measuring surface. 108-112. 215. 231 
Measuring temperature. 116. 117 
Measuring timp. 118. 119. 216 
Measuring value. 105. 106. 213, 214 
Measuring volume. 112, 113, 215, 231, 

277, 278, 312317 
Measuring weight. 114-116. 214. 231.232 
Measuring wood (not fire-wood), 8 
Median, 176 

Melting temperatures, 364, 367 
Members of equation, first and second. 

97 
Mensuration. See also measuring, etc., 

306-317 
Mensuration of circle, surface, 210, 211 
Mensuration of circumference, 204, 205 
Mensuration of cone (right circular), 

volume. 314 
Mensuration of cylinder (right circular), 

lateral surface, 313 
Mensuration of cylinder (right circular), 

volume. 313 
Mensuration of oblique parallelepip.ed, 

volume, 277, 278, 812 



GENERAL INDEX 



393 



Mensuration of oblique prism, volume, 
277, 278 

Mensuration of pail, capacity, 11 

Mensuration of parallelepiped (and fig- 
ure), (rectangular), volume, 58, 
112, 113, 312 

Mensuration of parallelepiped (oblique), 
volume. 277, 278, 312 

Mensuration of parallelogram, surface, 
110. Ill, 179 

Mensuration of prism (triangular) (and 
figure), volume, 277, 316 

Mensuration of prism (oblique), volume, 
277, 278 

Mensuration of prism (right), volume, 
277, 278 

Mensuration of prism (square), volume, 
112. 113, 277. 278, 312 

Mensuration of pyramid (and figure) 
(triangular), volume, 316 

Mensuration of ratio of circumference 
to diameter. 204, 205 

Mensuration of rectangle, surface, 31, 
32, 58, 108-110, 133, 134, 179 

Mensuration of rectangular parallele- 
piped, volume, 58, 112, 113, 312 

Mensuration of right circular cone, vol- 
ume, 314 

Mensuration of right circular cylinder, 
lateral surface, 313 

Mensuration of right circular cylinder, 
volume. 313 

if ensu ration of right prism, volume. 
277. 278 

Vlensuration of sphere, surface, 315 

ilensuration of sphere, volume, 317 

IdLensuratlon of square-cornered box, vol- 
ume, 57, 58, 112, 113. 

ilensuration of square prism, volume, 
112, 113, 277, 278, 312 

Mensuration of square, surface, 108, 179 

Alensuration of trapezoid, surface, 306, 
307 

Mensuration of triangle, surface. 111, 
179 

Mensuration of triangular prism (and 
figure), volume, 277, 316 

Mensuration of triangular pyramid (and 
figure), volume, 316 

Mercury, diameter of, 317 

Meridian, 297, 300, 304, 310 

Meridian, prime, 297, 300, 304 

Meridian, principal, 310 

Meteorology. See also weather, 242-244 

Meter ; also square and cubic meter, 
229-231 

Methods of shortening and checking cal- 
culations, 369-374 

Metre. See Meter 

Metric measures, 229-232 

Metric system and equivalents In United 
States system, 229-232 

Metric system, history of, 229, 230 

Metric ton, 232, 234 

Mile or knot, geographic. 227 

Mile ; also square mile, 214, 215 

Mile, statute. 214. 227 

Mileage and numbers of employees of 
street railroads,' 65 

Mileage of United States, 1900, rail- 
road, 816 



Milk, number of pounds In one gallon 
of, 204 

Milkings, weights of, 16, 75, 204, 241 

Mill, 213 

Milli-, 230 

Million. 18, and foot-note 

Minnesota quarries, 228 

Minus, 33 

Minuend, 33 

Minute of angle. 287, 289 

Minute of arc, 287, 289 

Minute of time, 217 

Mixed decimal, 193 

Mixed number, 151 

Model and development of cone (right 
circular), 314 

Model and development of cube (three- 
inch), 274 

Model and development of cylinder 
(right circular), 313 

Model and development of flat prism, 
276 

Model and development of oblique paral- 
lelogram prism, 277 

Model and development of prism (flat), 
276 

Model and development of prism (ob- 
lique parallelogram), 277 

Model nnd development of prism (right), 
276 

Model and development of prism 
(square), 275 

Model and development of prism (tri- 
angular), 277 

Model and development of pyramid (tri- 
angular), 316 

Model and development of right circular 
cone, 314 

Model and development of right circular 
cylinder, 313 

Model and development of right prism, 
276 

Model and development of roof. 307, 309 

Model and development of room, 109 

Model and development of square prism, 
275 

Model and development of three-inch 
cube. 274 

Model and development of tower, 309 

Model and development of triangular 

, prism, 277 

Model and development of triangular 
pyramid, 316 

Molasses into the United States, Imports 
of, 27 

Money, 105 

Money, kinds of, 213, 214 

Month, calendar, 217 

Moon, diameter of, 317 

Moon, mean distance from earth, 19, 
206. 331 

Moon, velocity of, 83 

Mountain time, 303 

Mountains, elevations and heights of, 37, 
69, 243. 247 

Movement of wind, 242, 243 

Multiple, 148, 149 

Multiple, common, 148 

Multiple, least common, 149-151 

Multiplicand, 44 

Multiplication by factors, 61, 52 



394 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Multiplication by fractional numbers. 54 

Multiplication by number near 10, 100, 
1000, etc., 52 

Multiplication by 10, 100. 1000, etc., 52 

Multiplication by 25, 50, 12 V^, 75. 500, 
250, pp. 52, 53 

Multiplication, checking, 53, 54, 369, 
371-373 

Multiplication compared, dirision and, 
61-63 

Multiplication (of common numbers), 
43-60 

Multiplication of decimals. Sec decimals 

Multiplication of fractions. Bee frac- 
tions 

Multiplication, short methods for, 51-53, 
192, 371-373 

Multiplication table, 45. 46 

Multiplication when some dibits of mul- 
tiplier are factors of others, 53 

Multiplier, 44 

Myria-, 230 

Name value of diffit, 17 

National League, one season's record of, 

238 
Nature study, 19.*). 106 
Negative, 116, 117, 175 
Neptune, diameter of, 317 
New England. 29 
N-gon, sum of angles of, 203 
Nines, casting out the, 28, 53, 54, 75 
Non-terminating decimal, 209 
Noon, 217 

North America, area of, 247 
Notation and numeration, 17-21 
Notation, Arabic, 20 
Notation, change of, 21 
Notation, decimal, 19, 190, 191 
Notation, English, 18, and note 
Notation, French, 18, and note 
Notation, German, 18, and note 
Notation of decimals, 190, 191 
Notation. Roman, 20, 21 
Notation, United States, 18, and note 
Note, promissory. 259, 260 
Notes, coupon, 345 
Notes, discounting, 260, 261 
Number, composite, 52. 143 
Number, denominate, 213 
Number, even, 143 
Number, mixed, 151 
Number, odd, 143 
Number of decimal places, 198 
Number, prime. 52, 143 
Number, square of, 321 
Numbers, common uses of, 127-134 
Numbers, compound denominate, 213- 

225 
Numbers, divisibility of, 73, 74 
Numbers for individual work, 41, 42 
Numbers, literal, 43 
Numbers, periods of, 18 
Numbers, reading. 19, 20 
Numbers, subtraction of literal, 42, 43 
Numbers, use of letters to represent, 

. 345-369 
Numliers, writing, 20 
Numeral, Arabic, 20, 21 
Numeral, Roman, 20, 21, 118, 119 



Numeration, notation and, 17-21 
Numeration of decimals, 191, 192 
Numerator, 139 
Numerical measure, 136 

Oblique parallelepiped, volume of, 277, 

278, 312 
Oblique prism (and figure), volume of, 

277, 278 
Observatories, longitudes of, 301 
Obtuse angle (and figure), 289 
Ocean depth, greatest, 69 
Octagon, sum of angles of, 293 
Octant, 286 

Odd number, 143 
One-brick wall, 78, foot-note 
Operations, order of, 159 
Opposite or vertical angles, 288 
Order of factors. 160 
Order of operations, 159 
Ounce, 214 
Ounce of gold, value of, 55, 115 

Pacific time. 303 

I*ail, capacity of. 11 

Taper-folding (creasing), 278-281. See 

also constructions 
Paper, measuring, 217 
Par, above, 341 
Par, at, 341 
Par, below. 341 
Parallel, 298 
Parallel lines, 180 
Parallel lines and line cutting them, 

angles formed by, 288 
Parallel lines, figure of, 99, 100, 177, 

275 276 288 
Parallel* ruler (and figure), 177, 178 
Parallelepiped, volume of oblique, 277, 

278, 312 

Parallelepiped, volume of rectangular, 

58. 112, 113, 312 
Parallelogram, 110, 283 

Altitude of. 111 

Area of, 110, 111, 179 

Base of, 111 

Compared (and figures), circle and. 
210 

Figure of, 110, 111, 179, 292, 203 

Properties of, 293 

Sum of angles of, 292 
Partial payments, 261-263 
Partial l»ayments, United States Rule 

for, 262 
Paving, 77, 78 
I»ayee, 259 

Payments, partial, 261-263 
Peck, 71, 216 

Pencil-foot of compasses, 98 
Pen-foot of compasses, 98 
Penny, 213 
Pennyweight, 214 
Pentagon (and figure), 282, 826 
Per annum. 252 
Per cent, 124, 235, 236 
Percentage, 123-125, 235-251 
Percentage and interest, 123-126, 235- 

263 
Percentage, applications of, 336-345 



GENERAL INDEX 



395 



Perch of stone, 205, 215 

Perlgon, 287, 289 

Perimeter, 105, 154, 282-284 

Periods of numbers, 18 

Perpendicular, 278 

Perpendicular bisector, (and figure), 

101, 176 
Personal property, 338 
Persons, heights and weights of, 13, 121, 

122. 106, 197, 206 
Pfennig, 214, 277 

Physical measurements, 11-14, 206, 207 
Pi, value of, 205, 370 
Pigeon, speed of carrier, 206 
Pin-foot of compasses, 98 
Pint, 216 

Piston, figure of, 211 
Place value of digit, 17 
Places, number of decimal. 198 
Planets, diameters of, 317 
Play-house, figure of, 183 
Plotted curves, 86, 120, 122, 267, 368 
Plotting observations and measurements, 

120-123 
Plus, 22 

Point (decimal). 190 
Point (geometric), 274 
Pointing off product of decimals, 198, 

199 
Policy, face of, 337 
Policy, Insurance, 337 
Population, density of, 70 
Population of Switzerland, area and, 69 
Population of United States, 1790-1900, 

123 
Population of United States, 1900, dis- 
tribution of. 29-31 
Populations of cities, 28, 41 
Populations of countries, areas and. 70 
Populations of states, territories, etc., 

1890, p. 39 
Populations of states, territories, etc., 

1900, p. 30 
Positive, 175 

Positive and negative readings of ther- 
mometer, 116, 117, 365, 367 
Pound, 214 
Pound, comparison of avoirdupois and 

troy, 116 
Pound sterling, 213 
Pounds in one gallon of milk, number of. 

204 
Power (of a number), 192 
Practical applications of proportion, 

188-190, 327-336 
Precloitatlon in various places, 25, 56- 

59, 193, 194, 244. See also weather 
Precipitation, inch, 57, 193 
Preferred stock, 342 
Premium, 337 
Present worth. 260 
Pressure of air. 127. 128 
Pressure of wind, 14, 15 
Price. Bee marking goods 
Prices of provisions. 4 
Prime factor. 142. 143 
Prime meridian, 297, 300, 304 
Prime number, 52. 143 
Prime to each other {phrase) ^ 142 
Principal, 248. 252 
Principal meriaiftn, 310 



Principles for using the equation. I 
350, II 350, III 351, IV 352, V 
352, VI 366 

Principles of decimals, I 193, II 199, III 
203 

Principles of fractions, I 141, II 144, 
III 146, IV 150, V 152, VI 152, 
VII 153, VIII 158, IX 166. X 170, 
XI 170 

Prism (and figure), volume of triangu- 
lar, 277, 316 

Prism, figure of oblique (oblique pile), 
278 

Prism, figure of right (square pile), 277 

Prism, model and development of fiat, 
276 

Prism, model and development of oblique 
parallelogram, 277 

Prism, model and development of right, 
276 

Prism, model and development of square, 
275 

Prism, model and development of trian- 
gular 277 

Prism, volume of oblique, 277, 278 

Prism, volume of right, 277, 278 

Prism, volume of square, 112, 113, 277, 
278, 312 

Prisms, comparison of, 276-278 

Problem, statement of, 44 

Problems of grocery clerk. 92 

Product, 44 

Products, table of, 45, 46 

Products of means and of extremes, 187 

Products of sums and differences of 
lines, 293-295 

Products, 1897, 1902, values of manu- 
factured and industrial, 40 

Promissory notes, 259, 260 

Proper fraction, 151 

Property, personal, 338 

Property, real. 338 

Proportion. 136, 137, 186-190. Bee also 
similar triangles 

Proportion, practical applications of, 
188-100, 327-336 

Proportion, terms of, 186 

Protractor (and figure), 286-289 

Provisions, prlcrs of, 4 

Pull. Bee tractive force 

Pulse, rapidity of. 51 

Pure decimal. 193 

Pyramid (and figure), volume of trian- 
gular, 316 

Pyramid, dimensions of Great. 816 

Pyramid, model and development of tri- 
angular, 316 

Q., 232 

Quadrant. 229. 285. 289 

Quadrilateral (and figure), 283, 306, 

307 
Quadrilateral sum of angles of, 292 
Quadrillion, 18. and foot-note 
Quarries. Minnesota, 228 
Quart, 216 
Quintal. 232 

Quintilllon, 18, and foot-note 
Quire, 217 
Quotations and transactions, stock and 

bond, 341 



396 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Quotient, 62 

Radical sign, 322, 327 

Radius, 98, 107 

Radius of earth, mean, 19, 316 

Radius of sun, mean, 19 

Rail, weight of one yard of steel, 54, 
64, 67 

Railroad cars, engines and tenders, 
heights, lengths, and weights of, 
15, 64, 129, 130, 266 

Railroad cars, tractive force for street, 
67, 68 

Railroad data, street, 65 

Railroad mileage of United States, 1900, 
316 

Railroad trains, problems on, 128-130 

Railroad trains, tractive force for, 129, 
130, 266-271 

Railroads, mileage and number of em- 
ployees of street, 65 

Rain and snowfall. See also precipita- 
tion, 193, 194 

Rain gauge, figure of, 56 

Rainfall. See also precipitation and 
weather, 56-59 

Rainfall, Inch, 57, 193 

Rains, numbers of, 25. See also precipi- 
tation 

Range (in land surveys), 310 

Range of temperature, 86 

Rate, 236, 252 

Rate of running, average, 132 

Rate per cent, 236, 252 

Ratio, 135, 136. See also similar tri- 
angles 

Ratio and as equal parts, fraction as, 
138-140 

Ratio of circumference to diameter, 204, 
205 

Reading numbers, 19, 20 

Reading of thermometer, 363 

Readings of thermometer, positive and 
negative, 365, 367 

Real property, 338 

Ream, 217 

Reaumur thermometer, 363-369 

Reciprocal, 170 

Rectangle (and figure), 110, 179, 282- 
284 

Rectangle, area of. 31, 32, 108, 179 

Rectangle, properties of, 293 

Rectangular parallelepiped (and figure), 
volume of, 58, 112. 113. 312 

Reduction, see also equivalents 

Reduction, ascending, 225 

Reduction, descending, 225 

Reduction of common fraction to deci- 
mal, 208, 209 

Reduction of decimal to common frac- 
tion, 192. 193 

Reduction of fractions, 140-142. 146, 
147, 151-153, 192. 193. 208. 209 

Reduction of fractions to higher, lower, 
and lowest terms, 140-142 

Relation signs, 350 

Remainder, 33 

Rent, cash, 10 

Rent, grain, 10 

Repeteod, 209 

Rhombus (and figure), 282, 283 



Rifle ball, velocity of, 206 

Right angle (and figure). 278, 289 

Right circular cone, volume of, 314 

Right circular cylinder, lateral surface 
of, 313 

Right circular cylinder, model and de- 
velopment of, 313 

Right circular cylinder, volume of, 313 

Right prism, model and development of, 
276 

Right prism (square pile), figure of, 277 

Right prism, volume of, 277, 278 

Right section, 211 

Right triangle (and figure), 179, 319, 
320 

Right triangle, relation of sides of, 320, 
321, 325 

River basins, areas of, 37, 247 

Rivers, lengths of, 247 

Rivers, rapidity of, 206 

Road wagon, tractive force for, 67, 199- 
202, 269, 270, 348 

Rod (unit) ; also square rod, 214, 215 

Roman notation, 20, 21 

Roman numeral. 20, 21, 118, 119 

Roof, figure of house and, 59 

Roof model and development of, 307 

Roofing and brick work, 307-309 

Roofing, square of, 307 

Room, model and development of, 109 

Root, short method for square, 374 

Root, square, 322-326 

Roots, cubes and cube, 326, 327 

Roots, squares and square, 321-326 

Ropes, data concerning, 360 

Ruble, 213, 214 

Ruler (and figure), parallel, 177, 178 

Running time, 131 

Rye, area and yield of Kentucky, 70 



Saturn, diameter of, 317 

Scale drawings, 1, 2, 7, 24, 76, 79 81, 

108, 119, 133, 134, 183-195, See 

also models and plotting curves 
Scale drawings of familiar objects, 183- 

186 
Scalene triangle (and figure), 103, 282 
Scales, figure of, 97 
Schedule, train, 131 
School data, Cuban, 26 
School data for largest cities of United 

States, 41 
School data for states, territories, etc., 

30, 37 
School grounds, figure of, 185 
School house and grounds, figure of, 

185, 186 
School room, figure of, 1 
Score. 217 
Scruple, 214 

Second of angle, 287, 289 
Second of arc, 287. 289 
Second of time, 217 
Section of land (and figure), 119, 215, 

311 
Section of land, figure of divided, 119, 

311 
Section, right, 211 
Sextant, 286 



GENERAL INDEX 



397 



Shape, likeness of, 137 

Share of stock, 341 

Shilling, 213 

Shingling, 307. 308 

Ships. See battleships 

Short division. 63. 64 

Short methods for addition, 360, 370 

Short methods for division, 70-74, 101, 

373 
Short methods for multiplication, 51- 

53, 192, 371-373 
Short methods for square root, 374 
Short methods for subtraction. 360-371 
Shortening and checkine: calculations, 

methods for, 369-374 
Sides of angle, 288 
Sides of equation, left and right, 97 
Sides of similar triangles, corresponding, 

328 
Signs, relation, 350 
Signs: 
' foot, feet; minute of angle and of 

arc, 79. 289 
" inch, inches ; second of angle and of 

arc, 70, 289 
-f plus, 22 

= equal, equals, 22, 97 
— minus, 33 

X times, multiplied b.v, 44 
-=- / — divided by, 62. 63 
% hundredth, hundredths. 124. 236 
<, > is less than : is greater than, 349 
± perpendicular to, 319 
V square root of. 322 
f cube root of, 327 
Silver coins, fineness of United States 

gold and, 55, 226 
Silver, value of one pound of, 55 
Similar triangles. See also ratio and 

proportion, 137. 327-329 
Similar triangles, uses of, 327-336 
Simple decimal, 193 
Simple Interest, 125, 126 
Six per cent method, 125. 126, 252 
Sleeping car, length of, 64 
Soil, 195 

Solar day, mean, 118, 217 
Solar year, mean, 217 
Solidus, 63 

Sound, velocity of. 55, 83. 206 
Sparrow, speed of. 206 
Specific gravity, 207, 208 
Sphere, area of surface of, 315 
Sphere, volume of, 317 
Spirometer (and figure). 11, 12 
Square (and figure), 282, 284 
Square, area of, 108, 179 
Square, center of, 320 
Square cornered box, volume of, 57, 58, 

112 
Square, inscribed, 182 
Square of number, 321 
Square of roofing. 307 
Square prism, model and development 

of, 275 
Square prism, volume of, 112, 113, 277, 

278, 312 
Square root, 322 
Square root bv subtraction, 874 
Square root» cnecking, 874 



Square root, short method for, 374 

Square roots, squares and, 321-326 

Square unit, 31, 108 

Squares and square roots, 321-326 

Squares of units, tens, and hundreds, 
table of, 322 

Standard base line, 310 

Standard time, 303-305 

Statement of the problem, 44 

Statements in words and in symbols, 
354-360 

States, territories, etc., areas of, 30. 35 

States, territories, etc., dates of admis- 
sion and populations (1890) of. 39 

States, territories, etc., populations 
(1890, 1900), of, 30, 39 

States, territories, etc., school data for, 
30 

Stature of persons. See aUo heights, 
etc., 196, 197 

Statute mile, 214, 227 

Steamboat, speed of, 206 

Steel rail, weight of one yard of, 54, 
64, 67 

Steel, weight of one cubic inch of, 113 

Stere, 232 

Stock and bond quotations and transac- 
tions, 341 

Stock certificate, 341 

Stock, common. 342 

Stock, preferred. .S42 

Stock, share of, 341 

Stocks and bonds, 341-343 

Stone, measuring, 205, 215 

Straight angle (and figure), 285 

Street railroad cars, tractive force for, 
67, 68 

Street railroad data, 65 

Street railroad mileage and numbers of 
employees, 65 

Streets and blocks, figure of city, 306 

Substituting, 267 

Subtraction, checking. 369, 371 

Subtraction compared, division and, 60, 
61 

Subtraction of angles, 290 

Subtraction (of common numbers), 32- 
43 

Subtraction of decimals. See also deci- 
mals, 195-197 

Subtraction of fractions. See fractions 

Subtraction of fractions with common 
fractional unit, 146 

Subtraction of literal numbers, 42. 43 

Subtraction, short methods for, 369, 371 

Subtrahend. 33 

Sugar production, 37 

Sum, 22' 

Sum and difference of angles. 289-293 

Sum of acute angles of right triangle, 
292 

Sum of angles of hexagon. 292 

Sum of angles of n-gon. 293 

Sum of angles of octagon, 293 

Sum of angles of parallelogram. 292 

Sum of angles of quadrilateral, 292 

Sum of angles of trianele. 291 

Sums and differences of lines, products 
of, 293-295 

Sun, diameter of, 317 



398 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Sun, mean distance from earth to, 19, 

206, 332 
Sun, mean radius of, 19 
S^un, rising and setting of, 223 
Sun time, local, 300, 305 
Supplemental angles, 291 
Surface measuring, 15, 31, 32, 108- 

112, 132-134, 215, 231, 306-312 
Surveying land, 310-312. 329-336 
Surveyors' measure, 215 
Switzerland, population and area of, 69 
Symbol. See sign 
Symbols, statements in words and in, 

354-360 



Tax, 338 

Taxes. See also assessment, 338, 339 

Tea into United States, imports of, 27, 

56, 203 
Telegraph wire, cost and weight of, 45, 

55 
Temperature. See also thermometer aft<I 

weather, 85-87, 244 
Temperature, measuring, 116, 117 
Temperature of a man's body, normal, 

368 
Temperature, range of, 86 
Temperatures, boiling, 116, 364, 368 
Temperatures, critical, 368 
Temperatures, freezing, 116, 364 
Temperatures, mean. 244, 364 
Temperatures, melting, 364, 367 
Temperatures of gases, critical, 368 
Ten-thousandth, 191 
Tenant farmer, 10 
Tenders, heights, lengths, and weights 

of railroad cars, engines, and, 15, 

64, 129, 130. 264 
Tenders, water and coal capacity of 

locomotive, 264 
Tenth (decimal), 191 
Terms of fraction. See also lowest 

terms, 139 
Terms of proportion, 186 
Territorial growth of United States, 246 
Territory of United States, area of ac- 
quired, 246 
Tests of divisibility of numbers, 73, 74 
Thermometer (and figure), centigrade. 

See also weather, 363-369 
Thermometer (and figure), Fahrenheit, 

85-87, 116, 363-369 
Thermometer (and figure), Reamur, 

363-369 
Thermometer, graphically, laws of, 368, 

369 
Thermometer, positive and negative 

readings of, 365. 367 
Thermometer, ransre of, 86 
Thermometer, reading of, 363 
Thermometer. See also weather. 85-87, 

116, 363-369 
Thermometers, equivalent readings of, 

367-369 
Thermometers, graduation of, 365, 366 
Thousand. 18 
Thousandth, 191 

Time belts of United States, map of, 303 
Time, central, 303 
Time, eastern, 303 



Time, equivalents of longitude and, 305 

Time, legal, 305 

Time, local (sun). 300, 305 

Time, longitude and, 300-305 

Time, measuring, 118, 119, 216 

Time, minute of, 217 

Time, mountain, 303 

Time, Pacific, 303 

Time, second of, 217 

Time, standard, 303-305 

Time, sun, 300, 305 

Time to run (phrase), 259 

To the weather (phrase), 307 

Ton, 214 

Ton, long, 214 

Ton, metric, 232 

Ton, number of cubic feet of hay and of 

coal in, 205 
Tonneau. 234 

Tower, model and development of, 309 
Town block and lots (and figure), 76 
Township (and figure), 120, 215, 310, 

311 
Tract of land, figure of, 7. 24, 119, 133, 

185, 306, 310, 311, 331-335 
Tractive force for railroad trains, 129, 

130, 266-268, 270, 271 
Tractive force for road wagon, 67, 199- 

202, 269, 270, 348 
Tractive force for street railroad cars, 

67, 68 
Trade discount, 249, 250. 339, 340 
Trade, 1891, 1901, United States export, 

40 
Trailers of locomotive, 264 
Train despatcher's report, 130-132 
Train schedule. 131 

Trains at Clhicago, number of daily, 63 
Trains, freight and passenger, 128-130 
Trains, problems on railroad, 128-130 
Trains, speed of express. 82, 83 
Trains, tractive force for railroad, 129, 

130, 266-271 
Transportation problems, applications 

to. See also tractive, force. 264- 

271 
Trapezium (and figure), 282 
Trapezoid (and figure), 184, 282, 283, 

306 

Altitude of, 306 

Area of. 306, 307 

Bases of, 306 

Dimensions of, 307 
Trees, lateral growth of, 147 
Trees, lateral growth of, 147 
Trees, leaves of. 195, 243 
Triangle, altitude of. Ill 
Triangle (and figure), equilateral, 102, 

103, 282, 284, 326 
Triangle (and figure), isosceles, 102, 

103, 282.. 284 
Triangle (and figure), right, 179 
Triangle (and figure), scalene. 103, 282 
Triangle, area of, 110, 111, 133, 134, 

179 
Triangle, base of isosceles, 102 
Trlansrle, relation of sides of right, 320, 

321. 325 
Triangle, sum of angles of. 291 
Triangles (and figures) and their -uses, 

similar, 137, 327-336 



GENERAL INDEX 



399 



Trlanfi:Ies (and figures), uses of the 30** 
and 60% and of the 46"', right, 180- 
182 

Triangular prism (and figure), volume 
of, 277, 316 

Triangular prism, model and develop- 
ment of, 277 

Triangular pyramid (and figure), vol- 
ume of, 316 

Triangular pyramid, model and develop- 
ment of, 316 

Trillion, 18, and footnote 

Trisect, 178 

Troy weight, 214 

True discount. 260 

Twilight, 224 

Unit, 18 

Unit, concrete, 213 

Unit, fractional. 107« 138 

Unit, square, 31, 108 

United States, land areas of divisions 
of, 201 

United States, maps of, 29, 303 

United States notation, 18, and note 

United States Rule for Partial Pay- 
ments, 262 

United States, territorial growth of, 
246 

United States, 1790-1900, population of, 
123 

United States, 1900, distribution of pop- 
ulation of, 29-31 

Uranus, diameter of, 317 

Use of letters to represent numbers, 
345-369 

Uses of numbers, common, 127-134 

Uses of similar triangles, 137. 327-336 

Uses of the equation, 848, 363-369 

Uses of the 30** and 60°, and the 
45% right triangles, 180-182 

Valuation, assessed, 338 

Value, face, 259 

Value, measuring, 105, 106, 213, 214 

Value of digit, name, 17 

Value of digit, place, 17 

Value of X, 44 

Values of cattle, weights and, 06 

Venus, diameter of, 317 

Vertex, 176, 274, 288 

Vertex of angle, 176, 288 

Vertical angles, opposite or, 288 

Vertices. Plural of vertex 

Volume, 277, 312 

Volume, measuring. 112. 113, 215, 231, 

277, 278, 312-317 
Volumes, 312-317 



Wagon, tractive force for road, 67, 199- 

202, 269. 270, 348 
Walls of house, 79 
Watch, figure of, 118 
W^ater area of earth, 246 
Water at greatest density, temperature 

of, 231 
W^ater. compared, weight of gold and, 

165 
Water, weight of one cubic foot of, 165, 

266 
Water, weight of one cubic Inch of, 164 
Weather. See barometer, heat, light, 

meteorology, precipitation, rainfall, 

temperature, thermometer, wind' 
Weather Bureau stations, elevations of, 

243 
Weather, to the (phrase), 307 
Week, 217 

Weight, apothecaries*, 214 
Weight, avoirdupois, 214 
Weight, growth of people, height and, 

13, 121, 122, 196, 197, 206 
Weight, measuring, 114-116, 214, 231, 

232 
Weight of one cubic toctt of varlouB 

things, 56. 64, 82, 114, 161. 165, 

207. 208, 235, 266, 313 
Weight of one cubic inch of steel, 113 
Weight to lung capacity, relation of, 

17 
Weight, troy, 214 
Well lighted (room), 81. 108 
Wheat yield and area of Dakotas, 70 
Wild duck, speed of, 206 
Winchester bushel, 216 
Wind movement. Bee also weather, 242, 

243 
Wind pressure, 14, 15 
WMnd, velocity of, 14, 15, 242 
Winds, names of, 14 
Wire, cost and weight of fence, 7, 8, 133 
Wire, cost and weight of telegraph, 45, 

55 
Wood (not fire-wood), measuring, 8 
World, map of, 296 
Worth, present, 260 
Writing numbers, 20 

Yard: also square and cubic yard, 214, 

215 
Yards In one mile, number, 54, note 
Year, 217 

Year, calendar, 217 
Year, common, 217 
Year, leap, 217 
Year, mean solar, 119, 217 

Zero, 17 



ANSWERS 

TO 
THE RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Page 6 

8. $.07 
4. $.18 
6. $2.75chanise 

9. $904.56 

10. ]\)th part 

Page 6 

11. $62.40 

12. 1897— $79.86 
1898— $80.40 
1899— $95.28 
1900- $95. 67 

1901— $101.59 

13. $21 . 73 

14. No. 
Breadstuffs, 

$6.49 

Meat $2.33 
Dairy and Gar- 
den $2.88 

Other Food .64 
ClothinK .90 
Metals 3.81 
Misc. 4 . 68 

IT.Bread $15.01 
Meat $8 . 04 
Dairy $13.67 
Other Food 

$9.01 

Clothing $15. 57 

Metals $14.54 

Mi»c. $14.73 

19. 

1897— 98— $.54 

1898— 99- $14.88 

1899— 1900— .39 

1900— 01— $5.92 

Page 7 

1. (a)160rods 

(b) i mile 

(0) 640 rods of 

wire 
3. 320 rods wire 

Page 8 

6. 640 posts 

7. $160 

8. $49.92 

9. $71.25 



Page 8 
10. $122.07 

18. 55 
13. 550 

19. $50.37 
16. 5: 10; 12 
17.20; 6; 7^; 16; 8; 

6f. 

18. 4; 8 

Page 9 

19. 6f: lOf 

20. 13i: 16; 24 

21. 6; 30 

22. $5.94 

23. $109.77 
24 $181.29 
25. $529.66 

Page 10 

1. $2,105.60; 
$13.16 

2. $8.16; 
$1,305.60 

3. $800 

4. for ^ of ail 
crops 

$192 more 
• profitable 
6. Corn more pro- 
fitable by $384 

6. Wheat $512 
more 

Corn $896 more 

7. Grain $160 
more 

8. Cash rent $96 
more 

9. 3264 bu. 

10. $1103.64 

11. $729.64; $374 

Page 11 

1. 121.70; 304.- 
25; 486.80 cu. 
in. 

2. 91.275; 106- 
487; 129.306; 
140.71 

3. 60.85; 7.606; 
64.653; 76.06; 
79.86; 83.668 

4. 121.70 cu. in. 



Page 14 

1. 35; 38901; 
11671* 

2. 662,500 lbs. 
8. 46681 lbs. 

Page 15 

4. 21787* lbs. 

5. 184078.125 lbs 

6. 154461; 1634,')i 

7. 49021.875 lbs. 

8. 1683 lbs. 

Page 16 

1.44.5; 47.8; 92.3 
2.N0V. 7.P.M. 2.9 

Page 18 

1.45.284; 6.934; 
321,113; 6,005 

2.9.010; 10.009; 
600,6«0; 5.050.- 
003; 80.080 

Page 20 

1.4.213,122.607; 
45,450,327.001 

2.718.200.000.- 
000; 190,402; 3,- 
000.124; 15.009; 
90; 186,000.000.- 
000; 147,100.100 

Page 23 

1. 242 

2. $12.10 
8. $1.85 

4. 35 

5. $.60 

6. 233 mi. 

7. 12 yd 

8. 21 yd.s. 

9. 19 yds. 

10. 44 spools. 

11. 4k doz. 

Page 24 

12. 34 

13. 19 lbs. 

14. 73 lbs. 

15. 265 lbs. 

16. 20 bu. 

17. 12 bu. 



Page 24 

18. 35 doz. 

19. 23 lbs. 
2. 1567 

Page 25 

8. 1031 

4. 66 

5. 287 

6. Chicago, 26 
10.46 in. 

Des Moines, 29 
11.33 in. 

Detroit, 32 
0.62 in. 
Kansas City, 28 
0.21 in. 

Omaha, 29 
10.12 in. 

St. Louis, 23 
11.29 in. 

Page 26 

7. 2876 ft. 

8. 3967 

9. 88,304 
10. 88,995 
11.3667 teachers 

and schools 
172.273 schol- 
ars 

12. Immigrants, 
1901—264,870 
male 

Immigrants, 
1901— 123.665 
female 

Total. 388,535 

Immigrants, 

1900—243,091 

male 

Immigrants 

1900—117,058 

female 

Total. 360,149 

Page 27 

13. Week No. 
1st.. 1,798,798 
2nd.. 1,823,561 
3rd., 1.814,340 
4th.. 1,820,085 
5th., 605.388 
Total, 7.862,172 



400 



ANSWERS 



401 



Page 27 

14. 148,890.876 lbs 
16. 245.083,133 

16. $61,180,966 

Page 28 

17. 2,050.000,000 

18. 9.477.398 

Page 29 

19. 17.436.267 

20. 26.^1 3.665 

21. 22,965,741 

1. 5,592.217 

2. 26.333,004 

3. 4.001,349 

4. 84,141.954 

Page 30 

6. 3.740.381 

Page 31 

6. 765.855 

7.620,290 N. Cen- 
tral 

8.Rhode Island 
1.250 Texas 
265.780 New 
York 7.268,894 
Nevada 42.335 
Nevada more 
than twice size 
of New Yock. 

9.956.731 5.823- 
019 

10. 15,341,220 

11. 28.686.008 
18. Population 

5.592.217; area, 
66,465 sq.mi. 
Population 15.- 
454,678 area 
102.200 sq.ini. 
14.956.731 ; 2,676,- 
509 

2. 50:5 

Page 32 

3. 6 

4. 9 

6. 6:15 

6. a 

7. 15z 
8.15 9's: 15 12's: 

15 x: 15z 
9.1. 17 a, II. 14 a, 
III. 73 a. IV. 
1462 z.V. 125 y, 
VI. 148 b, VII. 
23 m. VIII. 241 
n. IX. 197 c, X. 
191 d. XI. 1981 
X. XII. 1877 X. 

Page 33 

1. 238 mi. 

Page 34 

3. 408 

6. 437 

7. 7 yd. 

8. i pk. 



Page 34 

9. I yd. 

1.25i gal: 23 gal: 
18 gal: 15 gal: 
12^ gal: 10 gal. 

2. 7T)u. 

8. 15.000 

Page 36 
4. 1215 

6.(1) $7428.56 

(2) $2675.12 

(3) 299,189 yd. 
6.(1) 5,037,273: 

(2) $1282.78 (3) 
292i 
7.Depo8its $373.- 
80 

8. 30,466 

9. 3598 ft. 
10.68,526 sq. mi. 
11.47,259: 1955: 

201.649: 71.322 
12. 12,404: 254: 

24.058: 14.998 
18. 34.855: 1701: 

177,591: 56.324 
14.A11 less than 

previous week. 
Cattle, 15.929 less 
Calves, 1.859 less 
Hogs, 71.777 less 
Sheep. 33,206 less 

Corres. week 

1900 
Cattle, 6.706 less 
Calves. 133 more 
Hogs, 10,545 more 
Sheep, 16,276 more 

Corres. week. 

1899 
Cattle, 14,537 more 
Calves. 620 more 
Hogs. 41,925 more 
Sheep, 9,604 more 
16. All less than 
previous week. 
Cattle, 6,295 less 
Calves, 70 less 

Hogs. 5.428 less 
Sheep. 9.431 less 

Compared to 

corresp. week 

1900 
Cattle, 6,034 less 
Calves, 98 less 
Hogs, 483 more 
Sheep, 666 more 

Compared to 

Corresp. week 

1899 
Cattle, 2,848 more 
Calves, 44 more 
Hogs. 7,421 more 
Sheep, 11,816 more 
16.104,100 cattle 

393.000 hogs 

104.800 sheep 
17.Chicago great- 
est in cattle: 18,- 
000 more than 
Kansas City, 



Page 36 

30,500 more 
than Omaha, 
36.600 more 
than St. Louis. 
Chicago great- 
est in Hogs: 
112,600 more 
than Kansas 
City. 

136.400 more 
than Omaha. 
163,500 more 
than St. Louis. 
Chicago great- 
est in sheep : 
56,400 more 
than Kansas 
City. 

58,500 more 
than Omaha. 
65.500 more 
than St. Louis, 

Page 37 

18.As compared 
with the pre- 
vious day . 
Cattle deer. 40,500 
Hogs deer. 87,800 
Sheep deer. 68,800 
Corresp. week 
1900 
Cattle deer. 7.400 
Hogs incr. 54,100 
Sheep incr. 19,800 
Corresp. week 
1899 
Cattle incr. 12,400 
Hogs incr. 187,800 
Sheep incr. 23,.500 
Corresp. week 
1898 
Cattle deer. 24.300 
Hogs incr. 28,400 
Sheep incr. 9,200 
Corresp. week 
1897 
Cattle deer. 38,900 
Hogs incr. 19,800 
Sheep deer. 800 

1. 12,043 ft. 

2. 13,258 ft. 
8.1,250,000 sq.mi. 
4.43.950 sq. mi. 
6. Philippines 

smaller than 
Texas by 151.- 
370 sq. mi. 

Philippines 
larger than Illi- 
nois by 57.760 
sq. mi. 

Philippines 
larger than R. I 
by 113,160 sq.m 

6. 5,150.235 

7. 658,678T. 

8. $120,706,122 

9. 123 years 
10.264,530 sq. mi. 
11. 7,226,559 



Page 38 

12. 1,202,898 

14.So. Atlantic by 
113.870 sq. mi; 
No. Central by 
145,565 sq. mi. 
Western by 
567.245 sq. mi. 
16.No. A 1 1 a n- 
t i c, 10.603,415 
more; No. Cent. 
12.253.047 more 
So. Central 

9,988,608 more 
16.563,748; 1,191.- 
093; Maine ex- 
ceeds by 627,- 
345 
18.2,196,681; dif. 
in total pop. 
larger than thus 
by 250,663; dif. 
in total pop. be- 
ing 2,447,344 
20.Mainc 33,580 
N.H. 35,058 
Vt. 11,219 

Mass. 567.303 
R. I. 83,050 
Conn. 162,162 
N. Y. 1,271.041 
N. J. 438,736 
Penn. 1,044,101 
Total. 3,646,250 
24.W. N. Atlan- 
tic incr. 1,861.- 
506 over E. N. 
Atlantic.S. S. 
Atlantic incr. 
376,696 over N. 
S. Atlantic. E. 
N. Central incr. 
1,056.766 over 
W. N. Central. 
E. S. Central 
incr. 1679,088 
over W. S. Cen- 
tral. W. Wes- 
tern gained 155,- 
803 more than 
E. Western. 
W. Western 
gained 416,576 
more than 

MiddleWestern. 
E. Western 

gained 260,873 
more than 

Middle Western 
1. $1,871,568,787 
2.Un. Kingdom, 
+ $116,471,003 
Ger. +94,352,391 
Can., +65,809,640 
Neth. +54,382.038 
Mex., +21,400,198 
Italy, +19.599,197 
British Australia, 
+ 17,004,883 
British Africa, 
+ 21,483,098 



402 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Pase 38 

Jap., +17,323.093 
Bra., -3.928,245 
Arg. +9.207,733 
Ru8. +1,104.510 

Page 41 

S.Ag. Implements 

+ $1,832,143 

Books, maps, etc.. 

+ 517.837 
Carriages and cars 

+ 833,448 
Cop. ing. + 166.855 
Cot. clth. 

-1.114.683 
Cot.mfs. +650.981 
Cycles and parts of 

-241,087 
Bld'r'a hardware 

+ 357,616 
Swg. machines, 

+ 112,954 
Oth. Machinery, 

-328,378 
Corn, -302.141 
Wheat +1,220.799 
W. flour 

-1,777,158 
Coal, -1.614,679 
Cotton +1,882.526 
Fr. & n. + 778.676 
Furs and fur skins 

+ 471,630 
Cotton Seed Oil 
+ $214,619 
Beef, salted or pkl. 
+ 32.783 
Bacon, +192,408 
Hams, +30,879 



Page 42 

l.Pop. 8,074,559 
Pop. 5,974,091 
Sch. Enroll., 1,- 
213.519;no. tch. 
27,123; school 
ex.,$40.118, 540 

2.Pop., 1900, 

2,582.578; Pop., 
1890, 2.003.922; 
Sch. Enroll., 
367.287; no. 
teachers, 8.261; 
sch. expend . 
$9,756,432 
Pop., 1900, 

1,912.334; pop. 
1890, 1.496,982; 
sch. Enroll. 

310,325; sch. 
ex., $6,534,163 
no. teachers, 
7,356 

3. 120,415 

4.242; 6,261; 

267; 3,933 

5.9.084; 26,196; 
32,161 



Page 42 

6. $2,138,158; $- 
730,333; $654,- 
029; $525,965; 
$693,334 

7. More than 
half; less than 
half. 

S.Pop.. 1900. 13.- 
317.403; pop.. 
1890, 10,029,- 
377; sch. enroll. 
2,019.428; no. 
teachers, 45.- 
663; sch. ex., 
$60,176,405 
Incr. pop. 3,- 
288,026 

1. 4 

2. 7 
8. 6 9s 

4. 6 c's 

Page 48 

5. 9x, 18y; 6z; 
29z; 686a; 142a 

6. 120x aq. ft. 

7. 6x cts. 
8.6; 35; 95; 26; 8; 

5; 5; 6; 9; 12; 
2; 100; 32; lOi; 
72 

Page 44 

2. 126.720 

8. $2.04 

4. 768 

Page 46 

5. 258,300 

6. $20,664 
8. 512 
9.27,664: 1400: 

66x 
10.135,597: 231y: 
231 a 

Page 47 

1. Receipts, 
$512.18; Ex- 
penses. $128.23; 
profit. $383.95 
per 40 acres ; 

$9.^98 per acre 

Page 48 

2. Receipts. 
$296; expenses. 
$106.80; profit 
per 20 acres, 
$188.20; profit 
per acre, $9.41 

3. Receipts, 
$1803.20; exp.. 
$704.40; profit 
for 30 acres, 
$1098.80; pro- 
fit per acre, 
$36.62f 



Page 49 

4. Receipts, 

$638.76; exp., 
$166.69; profit 
60 acres, 

$472.06; profit 
per acre, 9.44/0 

6. Receipts, 

$114; expen. 
39.25; profit 
$74.76 for 10 
acres; profit per 
acre, $7,476 

6. Receipts, 

$368.00; ex., 
122.26; profit 
$228.15 for 10 
acre lot; profit 
$22,816 for 1 



Page 60 

7. Expendi- 
tures, $766.75 

8. Receipts, 
$3713.63; exp. 
$2034.37; pro- 
fit, $1679.16 
net earnings for 
the year 

1. 464.4 grains 

2. $0.80 
8. $2600 

4. $3.60 

5. $45.00 

6. $36 

7. 12 lbs. 

8. $4.20 

9. Milk; $1.63 

Page 51 

10. 4320; 103,680 

14. 16,200; 388.800 

15. 16,122.240,000 

16. $164.26 

Page 57 

1. 1 in.; 9 cu. in. 
2.9: 18: 45: 81 
3.144 cu. in: 288: 

432: 864 
4. 81 

6.^ inch, i, I. i. 

6. ' 144; 288; 72 

7. 12; 24; 24; 
SM; 936; 11,232; 

8424; 252.720; 
2106; 254,826 

Page 68 

10.373.33; 10,752; 

233,333i 
12.48 sq. in.; 336 

sq. in.; 2560 sq. 

in.; 182,250 sq. 

ft.; 9x; 9x.yd; 

ab mi., xy 
13. 27; 486 



Page 58 

14.81 cu. in.; 81 

cu. ft.; 81 cu. 

yds; 81 cu. rds.; 

81; 9a 
15.147 cu. in.; 96 

cu. in.; 06 cu. 

ft.; 180 cu. ft 

27 cu. yd.; 180; 

12x; 3xy; abo. 

17. 30.000 lbs. 
18,000 lbs. 

18. 10,330 lbs. 

Page 59 

19. 15,504 

20. $12.48 

21. 676; 108 
28. 10.944 lb.s. 
28. 20.736 cu. ft. 

1. 6; 8; 9; 23; 22; 
17; 8; 16; 9; 10; 
6; 2. 

2. a+b 
8. a-b 
4.ab; abc; xyz 
6.36; 96; 54; 72; 

288; 432; 1296 
6.8x; 7y; 25ab; 
axy 

Page 60 

7. ab; ex; 15abx 
1. 8 

Page 61 

7; 88 

9; 30 

30 

4 

13 

16 

15; 14; 16 

12; 13i; 20 



2. 
3. 
4. 
5. 
7. 
8. 
9. 
10. 



Page 68 



2. 
3. 
4. 
5. 
6. 
7. 
8. 



5 

30 

12 

$.90 

$.65 

$.27 

w. 14 yds, 

9. 30 mi. per hr. 

10. 12 gals. 

11. 25 

12. 84 

13. 55 

Page 64 

2. 1760; 3520 

3. 40 

4. 30 
6. 40 

6. 70 ft. 

7. 24 

8. 45 

9. 36 
10. 40.669§ per. 

mo.: 1355iVoper 
day 



I 

I 

:] 

:{ 

.'4- 
io; 



1 n. a 
i. ft;i: 

:9s 

7 w, ; 
. h; i 
;lNi)i., 
cu.yi. 
v: h: : 

m . 

10,11 



ANSWERS 



ik r 



i; ''" 
61 



7 

7 

20 

11 

6* 



Page 65 

11* 37.381 

12.46,693i; 51,- 

923J; 41,909f; 
^, 21,5441: 19,236 
IS. 20 

14. St. Joseph 5 

Memphis 

Oakland 

Hartford 

Worcester 

Peoria „_ 

16. 544; 786; 321; 
17.953; 7170; 
8411; 3071; 
3830; 112,836 

17. 8; 77; 700; 100; 
10; 9; 32; 15 

Page 66 

f- 79S 

*• 243 

5- 324 

5- 279 

I' 469 

;• 428 

iJ- 742 

10. 468 



403 



1. 

2. 

S. 

4. 

6. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 



Page 67 



2. 
S. 
4. 
6. 



Page 68 



Page 75 
i- 68 

;• 160 

i' 25 

"• 26.25 

Page 77 

1. 62,500 sq. ft. 

J. $2,000 

*• f 1,250 

* 6900 

11. |1.20; 618; $60 

13. $14.40; $4.80 

$72; $240 

Page 78 

J?*v r^ *219 

21.N. E. corner. 30 
ft. from E. line 
& 70 ft. from N. 
hne;N.W. cor- 
ner 34 ft. from 
W. hne & 70 ft. 
froni N. line; 
«A y*' corner, 
30 ft. from S. 
hne & 34 ft. 
from W. line 
*2. $112.60 

Page 79 

!• $5.60 

Page 80 

12,178 

$162 

^ $27 

$22.50 

$17.28 

$210 

Page 81 

$51.54 



Page 69 

6 nearly 7 

3 almost 4 

182.-f 

^49.6 

10.2+;1.58 + 

-w. 835.2 sq. mi. 
Jl. 34— nearly 130 
12. almost 270 
IS. 11.29 mi. 



1. 
2. 
4. 
5. 
7. 
9. 
10. 



14. 
15. 
16. 
17. 
18. 
19. 



Page 70 



147.8 , 
23.08 
16.7 j 
13 
^$589.23 + 
—' ^12,908.74 
20.269; 342; 235; 
140; 189; 293; 
15; 20.6 



Page 82 

1. 10Q 

2.iy?;12i;llj|; 
•• 143ii 

Page 83 

1901 

1100 

1188 

3.748,\ 

15271 
27 



Page 84 

!• 27^8? 

*• 42? 



Page 84 

|- 1275Jg 

8.8m;^$9454.8; 

»• 36381?: $20,- 
-« 493.18; $56.32 
10.6y5;$8516.23; 
$1229.72 

Page 85 

1. 7432?; $5352.- 
55; $71.54. 

Page 86 

i- 38; 38? 

f- 4U; .37,*. 

!• ?; 3?. 

4. g 

J- 12.75 

9. 78'';32<»;76* ' 

Page 88 
1. Adams, 49| 
Benson 
Boyd 
Claussen 
Denning 
Doan ,„ 

f-.. 28Hhrs 
S. Adams, M 

$2.25;T.$2.40; 
W. $2.40; Th 
$2.55; F. $2- 
70; S. $2.62i 
Benson, M 

$2.40; T. $2.55;* 



Page 88 

;• _ $84.45 

Boyd.^'''"^^' 
^- $16.20 



55;' S. ',2:70' 
Boyd, M. 0: T 
«1.20; W. $1.- 

$2.10 ' 
Claussen, M 
$2.40; T. $2.55; 
W. $2.55; Th 
$2.55; F. $2.70; 
S. $2.55 
Denning, M 
S2.55;f:$2.70;- 
W. $2.55; Th. 
$2.47i; F.$2.- 
70; S. $2.70 
Doan, M. $2.- 
40; T. $2.40- 
W. $2.40; Th! 
$2.40; F. $2.40 
|2.40; F. $2.40; ' 
^ S. $2.40 
4. Adams $14.92^ 
Benson 15.30 
Boyd 8.85 

Claussen 15.30 



8. Adams 
Benson 
Denning 
Uaussen 

9. Adams 
Benson 
Claussen 
Denning 

10. Adams 
Benson 
Claussen 
Denning 

^^Total 



11: 

3; 

4i; 

3 

1* 

3 

(3) 

4i 

$1.05 

1.80 

1.80 

^2.55 

$7.20 

18* 



Page 89 

*.pr. $2875.; Cr. 
J1232.75; On 
hand $1642.25 

Page 00 

5. Receipts $2.00 
gxpend. 1 . 25 
Balance $.75 

6. Receipts $105.- 
iS'* ..Expend. 
27.61. Balance, 

%i29.^7r-^"«' 
5- $17.50 

•• $5.86 



Page 91 

??• $48.75 

"• J9.78 

J J- $60.70 

"• $36.75 

Page 92 

"• $2.98 

1- 13c. 

?• 1 3c. 

J- 72c. 

*• 4.21c. 

**• 35c. 

g:e 93 

$3.47 
$276.01 
203.77 
173.13 
228.94 
$24,205 
29.00i 
27.195 
39.05 
18.55 
82.46 
881.85 
; $2.19; 
J $8.96; 
: $8.59 
$29.56 



404 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Page M 

S.S153.64;S89.32 
Page 96 

2.0ct. S168.37 
Nov. 173.16 
Dec. 155.82 
Jan. 166.93 
Feb. 148.94 
March 146.91 
April 110.24 
May US. 09 
June 120.49 
July 108.37 
August 118.29 
Sept. 111.51 
Total $1642.12 

S.Oct. $5.43 

Nov. 5.77 

Dec. 5.02 

Jan. 5.38 

Feb. 6.31 

Mar. 4.73 

4. $160.02 

6. $166.91 : $178.- 
63, 

Page 96 

S.April $3.87 
3.65 
4.02 
3.49 
3.65 
3.71 
$113.83 
$136.92 



May 
June 
July 
Aug. 
Sept. 



7. 
8. 



9.$575. 57; $64.92 

Page 97 

2.41b; y + 41b = 
141b. 

5. x = y 

4. 20. 

6. 35 

6. 41b. 

7. 4. 

8. 4. 

9. 3. 

10. 8. 

11. 3. 

Page 106 

1. 10; 20; 4; 100 

5. 2; 4; 5; 2^; U 

3. 21; 14; 7; 2 

4. *. 

6. 10 

6. 15; 3; 5; *; i 

7. $120; 24; 12; 

U; *. 

8. 1st; $400. 

9. 50:16. 

Page 106 

1. • 36c. 

6. 696 sq. ft. 

Page 109 

8. 64 sq. yd; 64. 

* 238 8q.yd. 

1228 sq.ft. 



9. 
10. 



Page 109 

11.2400 sq. ft.; 24. 

12. 1^; 213i sq. 
yds. 

Page 110 

13. 20; 30; 40; 80 

14. $48,400. 
16. $8,910. 

16. 18 sq. ft. 

17. 144 sq.ft. 

18. 120,000 sq. ft. 

60,000 sq. ft. 
60.000 sq. ft. 

19. 1080 sq. ft. 

Page 111 
28. ^ 

24. P-* ab 

28. 120.000; 120,- 
000; 70.000; 
60.000 

26. 50,000 sq.ft. 

29. 60,000; 60,000; 
50,000; 70,000; 
60,000 

80. 108 sq. ft. 

31. 120.000 sq.ft. 

82. 3128 sq. ft. 

88. 168 sq. ft. 

Page 112 

84. 1. 252 

2. 48 

3. 80 

4. 4860 

5. 640 

6. i mn rods 
Pq 



7. 



rods 



8. 2y 

1. 72 sq. in.;72sq. 
ft.; 60 sq. yds.; 
abc sq. yds. 

2. 1360 sq. ft. 
S. 1088 bu 

Page 113 

5. 355 

6. 64. 

7. 120. 

8. 10. 

9. 10. 
10. 3924. 
12. 567 lbs. 

15. 1782 1b. 
14. 180 cu. ft. 

16. 1346.49 + 

16. 264; U. 

17. 2176; U- 

18. 1080 cu. in., 
about i 

19. 1075 cu. in.; 
about i; 2160 
cu. in.; little 
over a bu. 



Page 114 

80 lbs; 2160 

81 lbs.; 2187 

125; 151 

4?; 27 

64; 3 A 

1U:1U 

4,',; 31118 

900; 1012^ 



Page 116 

10. 15c. 

11. 4,»,c. 

12. 88c. 

15. 88c. 

14. 17,982 lbs.; 
$2174.47 

16. 2,926 lbs.; 
$148.83 

16. 146.200; $8.- 
799.70 

17. 8.966 tons; 73.1 
tons 

18. $248.04 

Page 116 

19. $301,436 

20. $18,839 
22. 110.600 bu. 

1. 180* 

2. 30* 

5. 25* 
4. 16* 

6. 0* 

6. - 10* 

7. 15* 

Page 117 

8. 6*; 11*; 32*; 
:'2*; 14*: 9*; 9*; 
5i*; 3i* 

9. R5*:Rl^*;R3t** 
I'4^*;F18*;F6*; 
R8f*; R21i*; 
R5^°; F33i*; 
R4ri*; R90* 

10. R84*; R180*; 
R159*; F36*; 
F33*; F38*; 
F16*; Rlli*; 
F2U* 

11. +69* 

12. +9* 

15. -7* 
14. +3° 
16. +4*; -1*; 0*; 

-8* 

Page 118 

1. 2 

2, 12 hrs. 
S.i; 1: A; ,S; i 
4. Ihr. 

^•ih't A; ^i't 9^;Vti 

6. Ihr. 

7. 12 

8. Imin. 

9. a\i of \.'hole 
turn. 

10. 60; a second 

11. 24; 168; 720; 
8760 



Page 119 

12. 1440; 10.080; 

43.200; 625,600. 
IS. Hourhand VI.; 

Min. handXII.; 

Sec. hand 60 
14. Hour hand bet. 

V.andVII;mia 

hand bet. 48 

and 49; sec. 

hand at 46 

1. 102.400 sq. rd. 

2. 640 acres; 160 
acres. 

5. 160; 120; 80; 
80; 40; 40; 20. 

9. 2600 rd. 

10. 480 rd. 

Page 120 

11. ' 6ini.: 6mi.; 

12. 36 4sq. mi.; 23.- 
040 

14. 144 

18. 160x160 

Page 124 

6. .02; .06; .08; 
.12; .25; .33i; 
.87i; 1. 

^» A; A; «*»; Ai 

8.4lb.;'9lb.; 121b. 
9. 4; 12; 22; 100 

10. 2in. 

11. 12ft. 

17. $1 

18. 2in. 52in. 

19. 61.2 in. tall 



Page 126 



21. 
22. 

28. 
24. 
26. 
26. 
27. 
28. 
29. 



$1.25 

.05; S5 

$4.80 

$500. 

100* 

$9600. 

$50 



25%; 



76% 
25in. 

6 or 
/o 



Page 126 

1. $8 

2. .06 
S..03; .01; .005 
4. .015; .025; 

.035; .07; .08; 
.17 

6. 48c; $1.50; $2.- 
88; $6.10;$7.44; 
$27 

7. $54; $81; $121.- 
50; $330.75 

8. $1.68; $15.75; 
$26.25 

10. $48; $72; $120; 
$240; $4; $12; 
$8 



ANSWERS 



405 



Pace 126 

11. 6% 

13. $180; $198. 

14. $24.50; $70.87^ 
$85.75 

Pace 1S8 

S.Floor and Ceil- 
ing each 1,080. 
000 lbs. ea. end 
wall 432.0001bs 
ea. side wall 
540,000 lbs. 

4. 44^96. lbs. 

1. 39.6 mi. per hr. 

2. 3.10P.M 
$. 378 
4. 1 P. M 
6. 4301 miles 

6. 38 mi. per hr. 

7. 23imi. per hr.; 
18^. 

8. 41i; 34^ 

Pace 129 

9. 7 miles 

10. 50i mi. 

11. 13i 

12. 6 sec. 
18. i hr. 
14. 679 ft. 
16. 510.39 
16. 2.05195 
17.2712 ft.; 1329.- 

45 tons 

Pace ISO 

18. 2.82 T (omit- 
ting engine and 
tender) 

10. 3.192 tons 

20. 1.666 tons 

21. 2.51+ tons 

22. $156464.40 
28. 26.66i tons 
24. 11.75 tons 

Pace 131 

1. 45i; 50i; 47; 
46;49|:47i:45f 

2. 11; H; i; 2*; 
U; U 

3. A2i; B^; Ai; 
A2; B2; on 
time; Al. 

6. 55.5; 24.5; 

30.8; 50 
6. 50min.;20; 25; 

43. 

Pace 132 

7.1.11; 1.225; 1.- 
232; 1.165. 

8. 1.11; 1.07; 1.16 
1.20 

9. 73.5; 66.9: 
70.8: 72.4:66.9: 
70.5; 72.8. 

10. 38. 

11. 1070. 



PaC6 188 

1.88/6; $8401.56 

2. $50.40 

3. $63.35 

4. 18f acres;' 
$1406.25 

6. 37^ acres; 

$3281.25 

6. *; * 

7. $1500 

8. 1. 

9. 160; 220: 560 

Pace 184 

10. 3300 

11. $82.46 

12. 60. 

13. 151.875 ounces 

14. 4. If lbs. 
16. $5.62^; $7.05 

16. 2f; 45i; 160 

17. 40^; 324; 72. 

Pace 186 

1. ^; 2;^; 2;i;4; 
*;i:i;8;i;i; 



Aj 16. 
3!EtoF= 



;3;6:16 
= 2;Fto 

E =i;EtoG=- 

4; G to E -i; 

F to G -2; 

G to F -* 
4. 2; k; 3 

6. 3; i; 3; i 

6. 1;1;1;1;1. 
8. i; jV, i; iV, 3; 

12; 1 

Pace 186 

10. 16; 12; ,S; A; 
4* 32 

11. V; 320; 9; i 

12. 2; i;102. 400; 
160; 640 

18. 10; 20; |; !; i; 
2; 5 



2; 3; i 
Pace 137 

1. 10c. 

2. $.50 

3. 8c. 

4. 30c. 
6. 8 ft. 

6. 10 

7. 12 
1*8. 15; 24; 6 

9. 7; 7* 

10. 2; 8 

ll.A = 20; C = 42i 

Pace 138 

1. 2; i 

3. 2; 4; 6; 12; 3;9; 
18 

4. 2; 3; 6; 12; 4;8 
6. i 

8. 1 



Pace 138 

9. 4; 2 

10. ii 

11.1.3; 8: t; it; 13 
2. i: I: i; A; ii 
3f;|:3:.H; it 

Pace 139 

12. i; J;i;*;4;i; 

♦;i; A; A; A; 
13.4; i; i; A; A; 

17. i; ♦; A; A; A: 
ii«; A; A» 

1 . 1 

18. l;%;%^l',l;U 
f;i;8; i; I; a*.; 

19.^ 4 

20. 6; 8; 10; 12; 14; 
16; 18; 20; 22; 
50 

21. 1^; i)\ iJ\ 15: 18; 
21; 24; 27; :W; 
36; 45; *10; 90 

22. 10; l.l; 20: 25; 
30^ ^5; 40; 45; 
50; 100: 150 

23. il;a;S;t;S 

24. ,%: A; ,V, ^if^; 



Pace 140 

25. A; A; A; A;A; 

, ^ ill iV= «' 

1. 2; I; i''« 

2. 2; 3; A: A 
3.*; i; i; i;?;f 
4. 4; f; V, ? 

6. Same 

6. A: A A; U: IS; 

7. 2" 2 
8.'3;7;20;25;l'00 

Pace 141 

lO.A; \l; l«; 18; 
^^2I;U;A. 

14. I; I; I 

18.i: i; I: i; ?; ^ 



Pace 142 

2.1. 2, 3. 4. 6. 12 
3. 12 

4.'l, 3, 7, 21; 21 
6. 1, 2, 3, 6; 6 
6.1, 2,3, 4, 6, 12; 

12; 1; 2, 7, 14; 

14; 1, 2, 3, 4, 6, 

12; 12 



Pace 143 

8. a. 1. 2, 3. 5, 7. 

11. 13, 17, 19. 

23, 29, 31, 37. 
41. 43, 47. 53, 
59. 61. 67. 71. 
73. 79. 83. 89. 
97; 

b. 4. 6. 8. 9, 10, 

12, 14, 15, 16. 
18, 20, 21. 22. 

24, 25, 26, 27. 
28. 30, 32. 33. 
34. 35. 36, 38. 
39. 40, 42, 44, 
45. 46, 48, 49, 
50. 51. 52. 54. 
55. 56. 58. 60. 
62, 63. 64. 65. 
66. 68, 69, 70, 
72, 74, 76, 77, 
78. 80. 81, 82. 
84, 85. 86, 88, 
89. 90. fli, 92, 
03. 04. 0.5. 96, 
9S, 99, 100. 

10. 2. 4. 0, 8, 10, 
12. 14. IR, 18. 

20, 22. 24. 26. 
2H, :-;n, :v2, 34. 
3n, ;J8, Ai), 42. 

44, 46, 48, 50. 
1,:T, ,1,7,9. 11. 
1,^, Ifi, 17, 19. 

21, 2.S, 25. 27. 
2B. :\], 3.^, 35. 
.37, 39, 41, 43, 

45, 47,40, 
tlA. 8; 2; J6; 38; 

42; 

2. 5; 9; 21; 15; 
19; 27; 41; 

3. 5; 2; 19; 41; 

4. 8; 9: 21; 15; 
26; 27; 38; 42; 

12.9. 21; 8. 16; 3. 7 
14.7x3; 2x2x2x3; 

5x5; 3x3x3; 3x 

2x5 
16. 5. 

1. 2x2x3x3; 

2x2x2x2x3 
3. 2x3x9; 3x5x7; 

2x2x2x2x2x3 ; 

3x3x3x2x2x2; 

5x5x3; 5x5x5x2 



Pace 144 



8. 12; 14: 12 

9. 1. 20; 2. 18: 3. 
45; 4. 215; 5.36; 
6. 22; 7. 63; 8. 
163 

10. 1. g; 2. 2S; 3. 
."n; 4. S; 5. ,•; 
6. ih 7. n; 8. 
.?,; 9. I; 10. ?; 

11. A; 12. 4 



406 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



1. 

2. 

S. 

4. 

6. 

6. 

7. 

8. 

9. 
10. 
11. 

12. 
1 



Page 145 



Page 146 

1^: 



.. a: 
it; I'o'b 
Page 147 

i: I 

*./o;l:a; A; A; 
. A; W, 85; Si 

6J3 



^6^ 



70 
60 
12 
2 
20 
120; 



8. Uo"; 

Page 149 

2! 
6. 
7. 

8. 

11. 40; 24; 75 
360 

Page 161 

2. 36; 120; 180; 

1050; 168; 108 

4. g§i; igg; AV, 

Page 164 

8^8 
25,='i^ 
109iS 
Page 166 

William i ft. 

6. 518; 85S; i3/o; 

SOU; 413*3; 227 

8. 23iS 

9. 21 

Page 166 



4. 
6. 
6. 
7. 

2. 
3. 
4. 




Page 166 

14. ^la\, 

16! 65f; $1*1'. 7^ 

Ji- 
ll: '^^ 1 



Page 167 






$21. OH 
300; j\ 

32 
32 

V 

fi 



20. 
21. 
22. 
28. 
24. 

1. 

2. 

8. 

4. 

6. 

?: n 

8. 12; 4; 3; 6; 5; 
15; 4; 8; 6; 6; 
10; 10 

Page 168 

9. 21; 2j; Ig; 7i; 
3; f ; 3*; 2f ; 5i; 
31 ; 12^; 8i 

11. 5; 7; 7; 5; 19; 
16 



Page 169 



1. 

2. 

3. 

4. 

6. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 



3| 
231 



5i 

7\ 
6* 

$7 

16. 3f; $1.20; $6.00 

16. 5i 

17. li 

Page 160 

1^6; 20; 28; 10; 12 
18; 20; 16; 29}? 

2. 2Si; 231; 89i; 
231?; 233i: 

189i 

Page 161 

1. $7 

2. $70; $5 

3. 154; 192i 

4. 977* 
6. 357 

Page 162 

1. 15 sq. in.; 48; 
56 sq.ft. 160 sq. 
rda.; 320 sq. rds 
ab. 



Page 162 

2. 4 sq. ft.; 14 sq. 
ft.; 27 

3. V»5 sq. in. 

4. i sq. in.; A; *; 

A; A; A; §i; 
ii; Si 

Page 163 

Page 164 

8. 1277A 

t: «^l 

6. 1390^9 

7. 6I3JJ 

8. $4.12* 

9. $5,841 

10. $10,223 

11. $2,931^, 

12. $9354^9 

Page 166 

1.4; 5; 14; 16; 5; 

12; 8; c; m 
2.18; 32; 108; 325 

723; 14,665 

1- '^ 

i S 

6. 25c. 

Page 167 

5: . '^ 

9. ^e; $5,000 

'I. % 

Page 168 

8.135; MS; hV, 

Mli; m; 2?a; 

9A; Hi; 2A; 

8i 
4. 12*; 18.67?; 

16i; 12.67i1, 

Page 171 

1. $3i 

2. , i 

8. 39113 lbs 

4. 193 

6, 66 

6. 4f 

7. ; or 3* 

8. llg 

9. 3J 

10. 41 

11. 3* 

12. 10 
13.6g; 9?; 3|; 8V, 

m 

14.2U; 143; 9g; 

8 " 
16.9!'9J; 7*; 9A; 

6.^ J 



Page 172 

20. V 

21. $3100 

22. 1935 

23. 105 

Page 173 
J- ^* 

2- tSi 

8. 2Ji 

J- if 

6. la 

?: -^ 

8. 

9. 

10. 

11- 4^ 

J8. l.VA 

!*• li\iW 

16. ^ 

be 

1. 150 lbs. 

2. R. 11 lbs. 

Page 174 

8. R 3x 

4. 4 m lb. 

6. R 6* 

6.R 35X; R 51; R 
5115; L 45,»«; L 
65A; L 597A 

8:B 44,',; F 213^, 

TB7/4: F9/" 

Bl|8;B5X 

10.D4i4; U14S;D 

781; D 64=^ oa. 

Page 176 

12. + 3tlb8.; +ioz 
+ 33* T; +205- 
ilbs 

l.V/; tS«; i8: 
tkl; m; lAVo; 
121?; BU; 624; 

ia§; A; ir%; 

. 2|}V, a%; iig 
8.A; i4; 25; 1,VA 

%;3»A;A;iU; 

m; 141; O; 

10019; 10; 26}?; 

71ii 

Page 183 

1.4* ft.; 3f ft.; f>'; 

2r;2|';U'; 11' 
8.20'; 12*'; lo'; 

3*'; 7*'; 3i';6i' 
4. 298.'«sq. ft. 
6.75'; 33"; 60*; 2r 
7.33'x75«'; 30*'x7r 

6* above center 

and at sides; 

9*below 



ANSWERS 



407 



Page 184 

9. 32*; 48": 48" 
11. 9.492 sq ft. 
12.24" each; 28";2(r 
14.84"; 60"; 114"; 
84"; 9" and 3" 
16. 11; 10: 4^ 

Page 186 

1.320'; 400'; 470- 

156 sq. rds. 
2.200; 80; 58(V«»i 

5. 200; 160; 

4. 266i 

6.56'; 80'; 40'; 20' 

6. 50'; 85'; 80'; 
90'; 110'; 120'; 
170'; 210' 

7. 60'; 30': 15' 

8. 70,«oV9;i3,y,: 
6.61885: 20AV.; 

lll/o% 

10. 41rgcg sq. rds. 

Page 186 

11. 13012 rds. 

Page 187 

1.7*; 6; 18; 49; 4; 

2b; 6; 6; 2ab; a 
2.32; 15; 72; 88; 

4?; 24; 38; 2; 2; 

15; 25; 2; 2; i; 

i: 3i 



Page 188 

1. 75c. 

2. 60c. 

3. 90c. 

4. 12 
6. 60c. 

6. 30 

7. 2.40 

8. 18 
9.649; 663; 38.50 

10. 24; 32; 34 

11. 665. 96«, 

12. 28 

13. 29 

14. 2 ', 
16. 7607i 

Page 189 

16. 272,160 

17. 49 

19. 240; 50; 40 

20. 30 ft. 

21. 40 ft. 

22. 108 

22. 108 

23. 31 
24.9^; or; 3i";22i" 
26. 9" to mile 
26.1" to Sift.;!" to 

25': V' to Of 



Page 190 

27. Water, lOiozs.; 
Fat, 82; Prot., 
71; 6?; Water. 
18/oOzs.; Fat 
151; Prot., 12i'<»; 
12,V; Water, 
40JOZ8.; Fat, 
338; Prot. 31 i; 
261 

28. Water. 681ozs.; 
Fat, 201; Prot., 
2U; 17r'.: Wa- 
ter, 206|oz8.. 
Fat. 62?; Prot., 
64; 51 i; Water, 
258 ozs.; Fat. 
78; Prot.,80; 64 

29. Water, 15iozs., 
Fat, 5i; Prot., 
5: 4; Water. 73 
ozs.; Fat. 2t; 
Prot., 2i; 2; 
Water, 9/nOZ8.; 
Fat, 3/o; Prot.. 
2'g; 2i; Water. 
25iozs.; Fat, 
8-: Prot., 8; 63 

30. Water, 4iozs.; 
Fat, 2,',: Prot.. 
l/o; 2i; Water, 
lOgozs.; Fj 



Fat, 



5i; Prot.,4i;6i; 
Water, 6{t; 
Fat, 3?i; Prot.. 
2i|; 4; Water, 
32iozs.; Fat, 
16: Prot.. 12i; 
I8i 

Page 193 

2. 36.04 
3.1891 25.54 in. 

1892 36.56 

1893 27.47 

1894 27.47 

1895 32.89 

1896 33.14 

1897 25.85 

1898 33.77 

1899 31.49 

1900 28.65 

1901 24.52 
Total 327 . 35 

4.Jan.21.89 1.99 
Feb.25.15 2.88 
Mar.26.50 2.41 
Apr. 18.75 1.70 
May33.48 3.04 
June39.25 3.57 
July33 38 3.03 
Aug.29.04 2.64 
Sept.31.38 2.85 
Oct.14.54 1.32 
Nov.28.50 2.59 
Dec.25.49 2.32 
Total, 327.35 
1. 223. 76T 

2 157. 42i 

3. 127.35 

4. 1095i 



Page 193 

159 792 
1829.1 



Page 194 

2. 16.875 

3. . 133 lbs. 

4. .08171b. 
6.Wt. of water. 

.09751b.; org- 
anic .031 

6. Poplar water. 
.089; organic 
.0387; Ash wa- 
ter, .055; or- 
ganic .0414 

7. 1.0771bs 

Page 196 

8. 50.625 cu. in. 
1.04 1.48 

.04 2.71 

.06 3.51 

.08 2.85 

.10 3.20 

.12 3.16 

.15 7.36 

.26 12.13 

.31 15.04 

.33 17.16 

.34 20.56 

.34 18.61 

.45 16.10 

.41 16.31 

.35 17.67 

.35 18.74 

.35 18.73 

2.. 44 .44 

.40 .38 

.56 .54 

.36 .34 

.36 .34 

.35 .32 

.29 .18 

.08 .03 

.05 .03 

.03 .02 

.0 

-.03 -.14 

-.11 -.07 

-.06 

-.03 -.03 

.0 



Page 

. 6.37 

7.42 

11.15 

9.82 

16.04 

20.59 

20.16 

11.03 

4.87 

7.92 

.04 

-.46 

-3.89 

-4.80 

-3.73 





197 

5.14 

6.62 

11.81 

9.47 

16.08 

16.39 

15.39 

8.12 

2.75 

4.52 

. 1.99 

2.05 

-4.10 

-6.16 

-4.80 

•f.Ol 



Page 197 

4.2yrs.; 17; bet. 

40 and 50; bet. 

40 and 50 
6.2yrs.; 13; bet. 

40 and 50; bet. 

50 and 60 
7. Boys Girls 

.036 .040 
.052 -.020 

-.020 .080 

-.033 -.003 
.043 -.046 
.010 .039 

-.135 .053 
.132 .013 
.115 .020 
.549 .171 



Page 198 . 

1.3.811,808; 381.- 
180.8; 38.118.- 
08; 3,811.808; 
38.11808:3.811. 
808 
2.1; 2; 3; 2; 4; 4; 4 
3. 0; 1; 2; 3; 5; 3 



Page 199 

8. 143.547 

9. 806.7525 

1. Total wgt.. 
3236; H. puUiu 
lbs., 242.7; tot. 
wgt.. 3377.8; 
H. pull in lbs., 
253.335; total 
wgt.. 5256.35; 
H. pull in lbs., 
394.226i; total 
wgt., 6258: H. 
pull in lbs.,469.- 
35 

2. Total wgt., 
2240; H. pulfin 
lbs., 280; total 
wgt., 1709.58; 
H. pull in lbs., 
213.691; total 
wgt.. 5054.85; 
H. pull in lbs. 
631.856i; total 
w?t., 5335.75; 
H. pull in lbs., 
716.961 



Page 200 

8. 1060.9. 273.- 
7122; 4446.38. 
1147.16604 ; 
6669.25, 1720.- 
6665 



408 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Page SOO 

4.(1.) 7855.65, 
223.8860i; 149.- 
25735, 2ttl.- 
593145 

(2.) 11530.86, 
328.62951; 219.- 
08634, 383.- 
977638 

(3.) 12600.65 
359.118525: 
239.41235,419.- 
601645 

1. $62.75 

2. $62.75 

Page 201 

8. $37.25 

4.(1.) 18011.444 

18735.556 

62795.833 

67801.667 

106885.636 

(2.) 274230.136 

5. 1. 14.29 

2. 14.43 

3. 14.48 

4. 14.43 

5. 14.35 

6. 14.32 

7. 14.45 

8. 14.22 

9. 14.33 
10. 15.009 

Page 202 

6..1; .01; .001 

6. .5; .05 
7.. 10; 1; ,',; ^h^ 
8.. 01; .1; .01; 

.001 
9.6.8; .68; .068; 
68.; 680; 6.8; 

Page 203 
11.1. 23.8 

2. 2.38 

3. .00238 

4. 2380. 

5. 23800. 

6. 238. 

7. 2.38 

8. 23.8 

9. 23.8 
1.8122, 473, 954.- 

50 
2.46,073; 1355a\ 
8. $6,483 

4. 8298.70 

6. 88.785 

6. 823.126 

7. 8601.276 

Page 204 

8. 240.1; 201.9; 
442.10 

9. 812.85 

1. 3.144 

2. 3.138 



Page 204 

3.3.142; 3.151; 
3.142; 3.160; 
3.141; 3.193 
av. cir., 30.809; 
av. dia., 9.77 
ratio. 3.1546 

Page 205 

4. 3.119; 3.143; 

3.142; 3.142; 

3.142 
5.Av. cir., 134.3; 

av. dia.. 42.18; 

ratio. 3.139 
7.3.145; +.0034 

diff. 

Page 207 

1.18. 3.69 

19. 3.69 

20. 3.78 

21. 3.82 

22. 3.88 

23. 3.87 

24. 3.92 

25. 3.92 

26. 3.88 
2.18. 1.87 

19. 1.85 

20. 1.86 

21. 1.85 

22. 1.87 

23. 2.04 

24. 2.11 

25. 2.10 

26. 2.07 
8.18. 1.97 

19. 1.99 

20. 2.04 

21. 2.06 

22. 2.07 

23. 2.04 

24. 2.11 

25. 2.10 

26. 2.07 
l.Alum 2.664 

Zinc 6.984 
Oast Iron 7.2 

Tin 7.333 
Wrt.Iron 7.68 

Steel 7.840 

Brass 8.381 

Copper 8.832 

Silver 10.481 

Lead 11.350 

Gold 19.214 

Plat. 21.552 
Wood 

Cork .24 

Spruce . 5 
Pine.Yel. .554 

Cedar .561 
Pine,Wht. .448 

Walnut .670 

Maple . 750 

Ash .844 

Beech . 852 

Oak 1.04 

Ebony 1.216 



Page 207 

l.LignumV. 1.338 
Liquid 
Alcohol . 800 
Turpen. .870 
Petr. .891 

Olive Oil .912 
Lin. Oil .952 
8. Water 1.026 
Milk 1.032 

Acetic ac. 1 . 064 
Mur. acid 1 . 2 
Nitric ac. 1 . 52 
Sulph.ac. 1.842 
Mercury 14.08 



Page 208 

2. Gas 

Hydrogen .069 
Smoke 

(wd.) .090 

Smoke 

(B.C.) .101 

Smoke, 

212° .469 

Carb.ox. .967 
Nitrogen . 974 
Air 1. 

Oxygen 1 . 105 
Carb.ac. 1.529 
Chlorine 2.440 

8.Glass(av.) 2.81 
Chalk 2.79 

Marble 2.71 
Granite 2.66 
Stone 

(com.) 2.53 
Salt (com.) 2. 13 
Soil (com.) 1.99 
Clay 1 . 95 

Brick 1.89 

Sand 1.82 



Page 209 

2. .75; .40; .80; 
.875; .625; .187; 
.156; .203 

3. 1.50; 3.20; 2.- 
75; 6.875; 13.- 
312; 18.406 

4. . 14285; .23076; 
.47058; .39130; 
.43243; .26315 

6..333333;.666666 
.166666; .833- 
333; .111111; 
.222222; .777- 
777; .363636; 
.545454; .8181- 
81; .373737; .4- 
07407; .740740 

6. 1.35; 16.875; 
5.286; 17.9625; 
20.04; 1.008; 
.00875; .0875; 
.000625; 30.06- 
060 





Page 209 


7. 


.9166: .7692: 


.3684; .2727; 


4.2222; .4666; 


.8589; .7107: 


.5947 


8. 


.4702 + 


9. 


.301 


Page 210 


1. 


491.07^ 




122 . 767 




1257f 




11314? 


Page 211 


2. 


1413.72 


8. 


268.91074' 


4. 21480.945 lbs. 


6. 


Diam. 1.7506: 


area, 2.407 sq. 


in 


.' 


6. 7976. 71875 lbs. 


7.1. 


7951.4956 lb. 


2. 


47078.996 lb. 


3. 


31175.346 lb. 


4. 


37192.960 lb. 


5. 


42270.6 lb. 


6. 


31569.56 


7. 


73378.313 


8. 


a- ^ xr 




2 


9. 


C = 2HR 


10. 


763.41 sq. ft. 


Page 212 


1.1. 


99.89 


2. 


317.702 


3. 


1081.4 


4. 


366.731 


5. 


2917.7183 


2.1. 


78.58 


2. 


97.04 


3. 


12.93 


4. 


36.159 


5. 


37.896 


6. 


801.32 


7. 


723.037 


8. 


2.797 


9. 


25.252 


10. 


.017 


11. 


58.1211 


12. 


.0003 


3.1. 


3759.52 


2. 


677.8582 


3. 


756.0743 


4. 


8680.0868 


5. 


71.517 


6. 


.71517 


7. 


.06464 


8. 


1.209(5 


9. 


16.210801 


10. 


12.096 


11. 


12.093 


12. 


79.105488 


13. 


79.105488 


14. 


79105.488 


15. 


2184.1974 


16. 


. 21841974 


17. 


52.904544 


18. 


62.904544 



ANSWERS 



409 



Page 212 

4. 1. 4.5009 
2. 4.591 
3 88 
4! 320.152 

5. 28.4 

6. 38.6 

7. 4620. 

8. .0462 

9. 462. 

10. 462. 

11. 462. 

12. 231.000 

5. 1. 26.6661 

2. 876.388.888<i 

3. 1285.714 

4. 122222.22 

5. .333i 

6. .333i 

7. .666} 

8. .444j( 

9. 4888.888 

10. 4.333i 

11. .855K 

12. 12.312 

13. 3636.363 

14. 7575.757 
16. 78.787 

16. 7.878 

17. 3.463 

18. 11.441 

19. .191 

20. .636 

21. 112.222 

22. 33.333i 

23. 186 

24. I.IIU 



Page 220 

1. 230c.; 2300; 
23000 

2. $2.48; $24.80 
2480 

S. 49: 4.8 

4. 10 pounds; 
200 marks; 
250 francs; 
66f rubles; 

250 crowns; 

250 liras 

5.20.54 pounds; 

419.29 marks; 

518.13 francs; 

129.52 rubles; 

492.61 crowns; 

518.13 francs.. 
e, $7.9625 ' 

7. $323.08 

1. (1) 288; (2) 
1440; (3)4080; 
(4) 4080; (5) 
4608; (6) 26880 

2. 10 

5. 30 
4.1330 grains; 

2il troy ozs. 

6. 128 ounces; 96 
6. 384; 774 
7.4800; 22400; 

8008 



Pace 220 

8. (1) 56. (2) 300. 
(3) 1391. (4) 
1728, (5) 13,008, 
(6) 15,200 

9. (1) 2 tons; (2) 
l\ tons; (3) 8 
tons; (4) 4i 
tons; (5) 64 
tons; (6) 6* 
tons 

Page 221 

10. (1) 3 long tons 
(2) 8i 1. 1. (3) 5 
1. t. (4) 42i 1. t. 
(5) Vfl 1. 1. (6) If 

1. t. 
Il,5001b8.; I1201bs 

1. 198 inches; 
792 inches 

2. 66 ft. 

3. 678 inches; 
466 inches 

4. 32 rods. 2yd8. 
6. 60ft. 

6. -237.6 hds. 

7. 2840.2 bds. 

8. 90 

9. 336/r 

1. (1) 432 sq. in. 
(2) 440; (3) 
1.296; (4) 5.616; 
(5) 46.704; (6) 
6 588; (7) 4.636; 
(8) 6.516; (9) 
39«204 

2. (l)4sq.rds. (2) 
4 A. (3^ 6400 sq. 
rds. (4) i A. (5) 
272i sq. ft. ^6) 
4840 sq. yds. 

8. (1) 1 A. (2) 200 
sq. ch. (3) 6400 
sq. ch. (4) 230- 
40A. (5) 160 sq. 
rds. (6) 102,400 
Bq. rds. 

4. 2rdB.; 10 rds.; 
40rds.; 160 rds. 

, Page 222 

S.lOrds; 36sq rds 
more 
' 6. 60 sq. rds more 

7. 79 sq. yds less 

8. 43.560 

9. 201 sq.yds more 
1.1. 5pts. 

2. 10 gills 

3. 30 gills 

4. 16 pts. 

5. 48 qts. 

6. 94 pts. 
2.1. 12 qts. 

2. 6 qts. 

3. 9^ gals. 

4. 2 gals. 

5. 8qts.,lpt. 

6. 10 gal8.,2qts. 



Page 222 

3. 57^cu in.; 2S| 
cu. in. 

4. 37.8 cu. in 
6.1. 12pk8. 

2. 9pks. 

3. lOpks. 

4. 32dry gals 

5. 2ibu. 

6. 320pts. 

6. ,\|lb. 

7. 28bu.; 25bu. 

8. 806ibu. 32.256 
tons 

9.About i Win- 
chester bushels. 

10. 224pt8.;2ibu. 

11.31.08 per bu.; 
3|c. per qt. 

Page 228 

l.I. 3600 sec. 

2. 86,400 sec. 

3. 604,800 

4. ,iohr. 

5. 2hr8. 

6. 7da. 

7. 2wka. 

8. Iwk. Ihr. 5 
min. 52 sec. 

9. lO.OSOmin. 
2. 52 wks. Wed. 
6. $18171 
6. $2184 

9. 12:37hr8l9min 
12:66hr8 17 min 
13:13hrs 16 min 
13:29hrs 16min 
13:45 hrs 

Page 224 

10.6:14hr8.; 6:23 
hrs.; 9 min. 

11.6:26; 6:30; 4 
min.; 6:36; 6:37 
Imin.; 6:46; 6:- 
43; 3 min.; 6:45 
6:50; 5 min. 

13. Jan. Ist. 1:39 
April 1st. 1;35 
July 1st. 2:5 
Oct. 1st. 1:30 

14. Jan. Ist. 1:39 
April 1st. 1:34 
July Ist. 2:5 
Oct. Ist. 1:30 

1. $4.20 

2. 70; 2000 years; 
24000 months. 

8. 1776 

4. $24 

6. 129.60 

6. $432 

1. 15552 

2. 46A^ 

Page 226 

3. $25 

4. $37.56 
6. 1,244.160 
6. 2804 



Page 226 

7. 7rms, 7 quires; 
61rms., 9 qrs 

8. 560 shillings; 
6702; 26.880 

9. 216; 864 

10. 27,783 

11. 82.266 

12. £119. 6 s. 11 d. 
18. 234; 2808 
14. 2838 
16. 78yd8.; 2ft.;6in 

16. 19.200; 160 

17. 19,371 

18. 12cwt, lOlbs 

Page 226 

19. 5781in 
20.160yds. 1ft. 9in 

21. 277qts 

22. 55815 
28. 635 
24. $1925.92 
26. 37U 

26. 116.1 

27. 54.87 copper; 
18.29 nickel 

28. 232.2 

;;. 18H 

so. t; I: ^; 1 

81. 8.616 o8 

82. 1.324- 
88. 36 pounds; 

700 M 
84. 283.8 

86. I? 

36. 54ft. lOln. 

8/. 2mi., 44ch., Ird 

4li. 

Page 227 

88. 112rds., 4ydfl.. 
lOJin. 

89. 24.2 

40. 660 

41. 19881 

42. $975 

43. 5AA 

44. 10,000 
46. $192.50 

46. $14,400 

47. 6392.208sq rds 

48. 27.720; 16,'i 

49. 10. 

60. .37i 

61. $4.61 

62. $45 

Page 228 

63. 31.600 

64. 7.92 
66. $122828.28 

66. $530.30 

67. 24589.2 

68. $22.76 

69. $1.80 

60. 132^cu. in. 

61. $727.27 

62. 1188.46 

63. $2117.5 

64. $992.30 



410 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Pace 228 

66. lpt.a.376gill 

67. $13. S6 

68. 15S3 

Page 229 

69. Gain, 6.24 

71. .921875 

72. 478.12 

73. $76.65 

74. e'« 

75. 13ic. 

76. 2150.22 

77. 10.8 

78. $116.40 

Page 23S 

2.6m; 8dm; 7cm; 
5mm 

6. 2.112km 

7. $.711 

8. 48. 

9. 51 . 52m 

10. 4035m 

11. 10,000; 1,000- 
000. 

17. 215.28 sq. ft. 

Page 234 

20. 42m^ 

21. 262. 12m2 

22. $1664 

23. 372 ares 

24. 35m; 
26. 258.7846 
26. 2000; 2D1; .02 

Kl; .6804 qto 
Kl; .6804 qt. 
(liquid) 

30. 204.6211b 

31. 1000 

32. 13 Jibs.; 6000g. 

33. 39.81456 

34. 220.46; 
1102.31 

35. 38.58 capsu 

36. 32.15 cents 

37. $39.41 

38. $17 

39. 56 

Page 235 

40. 31.25 

41. 131. 7ei 1726 

Page 236 

1. 24 

Page 237 

2. 60% 

3. 40% 
4.. 834; .62i; 

.911; .14?; .71f 
.44*; .63/ 
.46^',: 
.485;; 






Page 237 

5. 481!%. 

7.334%; 25%.; 



r.334%; 25' 
4lf%; 77r 
97i%; 21% 

7iP!^ 



; I. 



8.1. 
2. 
3. 
4. 

6. 
6. 
7. 
8. 
9. 

9. A; ,\; 

10.. 07; .0625; .R;^ 
.875; .015; 
.0275; .STj; 
.001; .0001 

11. 11.27CU. hi 

12. 4% 



Page 238 

13. 64.7; 59.2: 
58.0; 64.2: 5U.0 
38; 37.9; 37 4 

14.25%;.12i;,[Xil 
.18i; 18i; .OrtJ 

15. I9arx» 

16. tH\ 

17. $2000 

18. $21W> 



Page 239 

1. $4;$7 .^n 

2. .02a; .OJt.; 
.02x; .12x 

3. .05a; ^or^h, 
.05x; .05z; .40/. 

4. .125a; .12o%; 
1.25X 

6. .Oa; Ox; .OOs: 
.12y; .45z 

7. .16; .29; n^i; 
ox; om; . J*'n 

8. 1.6r;3.5r;()f3i; 
.OrR; f'rr, 
.02rar 

9. p^ I'^r 



Page 240 



SJU 



1. 

2. 2av. 

3. Vi(^. 

4. $ 80 

5. IfiO 

6. $128; $179.20: 
$224; $40; im; 
$560 

7. $,^KJ2 

8. ' ■$440'> 

9. SOhf 



Page 241 

10. 1775% 

12: 27 A % 

13. 66§% 

14. 11.200 

15. $52.80 

16. $14.00 

17. 20% 

19. 32.000 

20. 697 

21. 2000 

22. $9,775 

23. $750 

24. $250 
26. $29.76 
26. $1 . 16i 



Page 242 



27. $5,000 

28. 15% spilled; 
$4.5 

29. $925 

30. 20%, 

1. 233't; 483?; 
22;f; 293't 

2. Feb. 35?; 39?; 
25; 324 

Mar. 38g?; 45^; 
16,^; 35Jt 
Apr. 33J; 43i; 
23i; 43i 

3. 213?; 24!?; 
29H: 24i? 

4. Cloudy. 28J5; 
20IS; 261!; 
242S 

P. Cloudy, 261!; 

261!; 19il,; 

23i! 

Rainy or S.21?; 

21?; 23,S; 30?? 

5.Jan. 71?% 

Feb. 68)%, 

March 81?% 
Apr. 100% 

May 77»% 

June 64if% 

July 80% 

Aug. 67;%, 

Sept. 74?%, 

Oct. 82;% 

Nov. 74?%, 

Dec. 82;%, 

6.Jan. 82%; 

Feb. 66% 

Mar. 90% 

Apr. 83% 

May 80% 

June 70% 

July 73% 

Aug. 67%, 

Sept. 76% 

Oct 84.9 or 85% 
Nov57.7or 58% 



Page 243 

7.Mt.T. 88.9% 
BIk. Isl. 94.9% 
Chicago 100% 
Cleve. 112.9% 
N.Y. 114% 
Buffalo 1 16% 
Boston 147% 
Phila. 152%, 
St.L. 171.7% 
New Or. 195%, 
Louis. 206% 
Wash. . .228% 
Roseb. 476.4% 

8.Mt.T. 34.6% 
Blk. Isl. 3165% 
Chicago 100% 
Cleve. 108% 
N.V. 2fi2% 
Buff. 1.07% 
Boston 658.4% 
Phila. 703% 
St. L. 145% 
N. Or. 1613.7% 
Louis. 156.7% 
Wash. 734.8% 
Roseb. 158.8% 

9. 39.5%; 2.4% 

10. 23: 1.6 

11. 120 sq. ft. 



Page 244 

12.Jan. def. 46.8% 

Feb. ex. 43.6% 

Mar. def. 33.7% 

Apr. ex. 38 % 
1. 60.2% 

2.St. Louis. 9 hrs. 

29 min.. and 

51 . 72 sec. 

St. Paul. 8hr.s.. 

49 min.. and 

14.16 sec. 
3.St. Louis. 14hrs 

51 min.. and 

52.14 sec. 

St. Paul. 15hra.. 

31 min., and 

2.52 sec. 



Page 245 

hrs min pcc 
5.F. 9 52 55.2 
M. 11 11 25.62 
A. 12 36 31.32 
M. 13 57 46.44 
J. 14 55 58.08 
J. 14 57 54.9 
A. 14 28 52. 2S 
S. 13 13 51.24 
O. 11 49 18.48 
N. 12 7 25.50 
D. g 23 16.44 

3. 6.5%; 89% 

4. 27.6%; 5.2% 



ANSWERS 



411 



Pace 246 

7. 21% 

9.L.Pur 142.8% 
Florida 2.9% 
Texas 17.9% 
Mex pur 21.4% 
Tex. pur 3.2% 
Gads, pur 1.4% 
Alaska 19.02% 
Hawaii . 18% 
P.Rico .09% 
Phil Is. 3.07% 
Guam .006% 
Is. of P. .02% 
10. 73.4%; 26.6% 



Page 247 

U.N.A. 43.1% 
S.A. 66.4% 
Asia 44.7% 
Africa 83.5% 
Europe 38.5% 
Aua. 77.1% 

12.N.A. 725 

S. A. 336i*r 

Asia 79^S 

Africa 438? 

Europe 350 
Austral. 350 

13. 48.7%; 72.5%; 
69%; 98.5% 

16. 7.7+; 2.2; 1.- 
7+; 1.4 + 

16. 47.6; 33i; 47.6; 
47.6 

Page 248 

17. 92.8%; 104% 

19. 71.9% 

20. 96.6%; 86.8% 

1. 64.00 

2. 162.50 
8. 61. 50 
4.1. 6100 

2. 6382.50 

3. 63.00 

4. 614.25 

5. 63 

6. 6112.50 

7. 675 

8. 6240 

Page 249 

6.646.25; 664.75; 
6231.25 

6. 62706.12 

7. 61119.52 

8. 613.684 

9. 653.6085 

10. 617.523 

11. 63886.40 

Page 260 

2. 660 

3. 62992.50 note 
62975 

4. 65301 



Page 260 

6.1. 630.375 

2. 6128.25 

3. 61054.62 

4. 41.2965 
6. 233.284 

6. 6686 

7. 617.859 
6. 24% 



Page 261 



1.1. 

2. 
3. 
4. 
5. 

2.1. 



2. 



ao 

3. Jlf2; 4. 
io 

5. ^; 6. 
has 

7. ^^ 
mos 

3.1. Hi!; 2. 



429 
41U 

28r* 
46 
26i' 
on . 
m s' 
hha. 



-3.. 



rho 

rns 



1; 4. 



has; 



;6. 



zus 
ais umo 

4.1. oo; 2. ee; 3. 

aah; 4. ren; 

5. Z2s; 6. hhrr 
5.1. mo; 2. io; 3. 

hao; 4. hzr; 5. 

rio; 6. mao 
e.l.k k; 2.ss; 3. 

lab; 4. aoh; 

5. rre; 6. bba 

lab; 4. aoh; 

bbaa; 

1. ck; 2. ok; 3. 

blk; 4. bra; 

6.aok; 6.cck 



8.1. uu ; 

r 
zs; .3. hns; 

rrr ; 6. 



Page 263 

1.631.50:631.50; 
663; 6157.50; 
6110.25; 6149.- 
625; 631. 50t 

2. 6100; 699.20: 
6297.60; 6678.- 
661; 699. 20x 

3. 9128 



Page 264 

6. 662.30 

6. 6332.4375 

7.1. 634.125 

Amt. 6734.126 

2. 633.90 
Amt. 6433.90 

3. 631.43 
Amt. 6242.43 

4. 620.504 
Amt. 6170.504 

5. 619.565 
Amt. 6299.565 

6. 619.40 
Amt. 6379.40 

7. 620.75 
Amt. 6280.76 

8. 668. 44$ 
Amt. 6568. 44$ 

9. 639.25 
Amt. 6339.26 

10. 6142.50 

Amt. 6767.50 

8. 6100. 

9. 6150. 

10. 670.44 

11. 6240. 

12. 6860. 



Page 266 
16.1. P = 6200 

A = 6219.50 

2. P = 6640.22 
A = 6809.24 

3. P = 6720.00 
1 = 6278.50 

4. P = 6135 
1 = 663.42 

5. P = 61200. 98 
A = 61323. 48 

6 P = 63424.864 
A = 63600.86 

7. P = 62888.88 
A = 2921.387 

16. 34?% 

17. Il3h% 

18. 12% 

19. S9^i 

21. (1) 5.6 + ; (2) 
10.1: (3) 11.9; 

(4) 10.4 

22. 4; yrs.; 3 mo. 
15 da. 



Page 266 

23. 5 yrs., 9 mo. 

24. 1 yr., 6 mo., 
15+ days 

26. 12 yrs & 6mo. 

26. 16fyrs.; 14?yr8 

27. 6yrs, Imo., 

16^+ a», 



Page 256 

30.1.6yr8, 2mo, 8^ 

or 9 da. 

2.1yr, Imo, 15 

da. 

3.3yrs, lOmo., 

27dfa. 

4.3yrs, 4mo, 18 

da. 

6.3yrs, 3mo, 6 

da. 
31.1.1,611.78; A = 

671.78 

2.1, 641.927; A 

= 6216.927 

3.T, 2yr.,3mo; 

1=6126 

4.T, 3yrs, 7mo. 

15da; A = 6617. 

50 

6.T, lyr, 7mo, 

26da; 1 = 83.85 

6.P, 61500; A = 

2009.25 

7.1, 687.161 

8.T, Syr, 7mo, 

6da; A = 6937.- 

986 

9.R, 8.9% 



Page 267 

1.1. 637.50 

2. 675.00 

3. 6243.75 

4. 6137.60 
6. 6968.75 

6. 637. 60x 

7. 37.50t 
2.1. 662.60; 2. 

6206.25:3.6525 
4.6250; 5.6518.- 
75: 6.637. 50t.; 
7.612.50rt.; 8. 
612.50 ru; 9. 
612.50 ru. 



Page 260 

4.1. Int., 642.04; 
Amt., 6292.04 
2.1nt, 6162.365; 
Amt, 6797.365 
3.1nt., 6537.53; 
Amt., 62937.53 
4.1nt., 6330.46: 
Amt., 64195.46 
5.1nt., 6771.83: 
Amt., 64411.83 

1. 6109.09 

2. 6132:622.91 



Page 261 

8. 666.79 

4. 661.47 

9f H547I W.40 



412 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Page 261 
6.1. D. $5.90; 
Pro. $59.41 

2. D, $31.21; 
Pro., $284 

3. D. $46.70; 
Pro., $462.36 

4. D, $123.12; 
Pro., $730.88 

5. D, $229.77; 
Pro., $810.42 

6. D, $376.48; 
Pro., $1167.64 

7. D, $360.31; 
Pro.. $1862.19 

8. D, $193.99; 
Pr... $3920. 137 

9. D, $157,438; 
Pro., $3920.137 

10. D,$1101.876 
Pro., $875687 

Pace 263 

2. $460,676 

3. $1920.669 

Page 266 

1. 4 

2. 9'5'' 
3.1. 85*' 

2. 61 r 

3. 87' 

4. 102* 

5. 137' 

6. 65' 

7. 57'5r 

8. 27'lir 

4. 102* to 1' 
6. 631' 

6. 68ft. fin. 

7. 140iinches 

8. 191.2 inches 

9. 127Vinches; 
89i inches; 140i 
inches 

10. 25i X 25i 

11. 3'; 49*; 27' 
Page 266 

12. 18'; ll3f 

13. RI9'; C119? 

14. 264; 169? 

15. 60mi an hr 

16. 2i times 

17. 181535 lbs. 

18. 121.333*lbs. 

19. l,080,868i lbs., 
or 540.434iton8 

20. 2522.026^1b8. 

21. 2972.387H lbs. 
Page 267 

22.La422.7tVil.;lb8. 
2.3H7:*.]n V:^lbs. 
^{,4H2hV47:M lbs. 
4.4773,8:lH->-:-.:^bs. 

ti.r»f574.5.i8^ lbs. 

7AM2'lM'2Uf: iba. 

8.r>-')7'i .'■<- ^bs. 

9,711;; ■■ I' ! 13 'bs. 

lU 792«. 31375 
12. 8370.T2ai', 



Page 269 



1.1. 

2. 

3. 

4. 

5. 
2.1. 

2. 

3. 

4. 

5. 
3.1. 

2. 

3. 

4. 

5. 
5.1. 

2. 

3. 

4. 

5. 



2161b8. 

3201b8. 

• 6621h8. 

817lb8. 

1223ilb8. 

85lbs. 

2031b8. 

2701bs. 

4801bs. 

572*lb8. 

97.501b8. 

1551b8. 

1171b8. 

4251bs. 

212.51b8. 

10401bs. 

12801bs. 

26501b8. 

38801b8. 

45001bs. 



Page 270 

9. 1. 20; 2. 20; 3. 

3; 4. 40; 5. 7000 

6. 40^1 + ; 7, 

60; 8. 12 

10.1. 1785.71lb8; 

2. 4821.42lb8.: 

3. 2010lbs.; 

4. 5910.711bs.; 

5. 2449.291b8.; 

6. 39410.71b8. 



Page 271 



11. 



L = 35.71F 

2. L = 66iF 

3. L = 30.30F 

4. L = 52.63F 

5. L = 83 33iF 

6. L = 52.63F 

7. L=125F 

8. L = 100F 
12.1. 9998.801bs. 

2. 5177.95lbs. 

3. 2806.806lbs.; 

4. 6277.818lbs. 

4. 6277.8181bs; 

5. 4999.401b8. 

6. 24,997lb8. 
13.1. 1999.801bs. 

2. 10.998. 90Ibs: 

3. 3999.60lbs. 

4. 15,998. 401b8; 

5. 799.921b8. 

6. 1599.841b8. 
14.1. lO.OOOlbs. 

2. 50.0001b8. 

3. 60,000lb8. 

4. 40.0001bs. 

5. 70.0001b8. 

6. 72401b8. 
15. (1) 200; (2) 

300; (3) 149.69; 
(4) 90; (5) 180; 
(6) 54; (7) 1620; 
(8) 32.4; (9) 
1100 



Page 271 

17.1. JsjHjnibs.; 

2. •^l,*Mni\h:.\ 

3. ^UMfMi^lliti.; 

4. ai^HKl; 

5. 2i^JiMii; 

6. :i2Ai>iK 

7. j:V-^N(h 

8. ^5,fjS0' 

9. .u).:^m 

19.1. lU,'j.{>00tbH.' 

2. ll4,00Cthft.: 

3. CMJ.OOOtba.; 

4. 122,4001be.; 
6. I41.000lbs; 
6. 14S.e5fJ 

20.1. 443:£ 

2. 232ft, 80 

3. 4.flHfi 

4. 9,0025 
6. . -inSfi 
6. J. 2021 5 

Page 29& 

18. 66S24 t<n. ft, 

19. 741 SCI. rr. 

20. 723 »<i. rr, 

21. 172,^^ at I. rii^ 
22.168/i\ sq. rti^.; 

1631J? 

23. 4U\9fi.[i. 

24. lAA 

25. 2^Ae 4|fA 

26. S|A 

27. 8A; 0l7f aq.rd^. 

Pagi 29a 

1.1. r,frw- 

2. 90* W; WSi 

3. 30* K: ;<(rN; 

4. 90'W. ;i,=^°:v I 

2. l. + 7/t°. +40^i 

2. +7^l^ +40°; 

3. +7fr\ +4.i^^ 

4. +90^. +^45^; 

5. +105°, +40: 

6. +30°, +»0; 

7. A'h f 52* 

8. -y, +50" 

9. +15^ +52^ 

10. +2rf, +80" 

11. +68", -55" 

12. -5^. +3^^^ 

3. 65°; 72*; 30^i 

4. 46°; 70^: 50^ 

5. Hfr^ SO" 

6. 36*; 22*: :<7* 

7. 35'*: 1 5": 30" 

8. 30"; 60°: 45° 

Page 399 

9. 45" 

10. 15^ 

11. 16"; :45^ (iri*^; 
near AH"" 

12. 30°; uear 50*; 
15* 

13. 15**; npar 35''; 
15*; 30" 



Page 299 

14. about same; 

same; 10*; 7* 

16. 16.48; 35.42 

16. dif. in dif. in 

lat. long. 

1 2.28 262.01 

2 70.78 50.80 

3 21.53 91.09 

4 6.05 63.04 

5 93.26 36.52 

6 5.67 200.62 

7 42.00 7.89 

8 20.83 18.58 

9 37.80 8.20 

Page 800 

2. 7o'c;12o'c 
8. 60*,46' 

Page 801 

4. 7hr., 3miTi. 
5.1 St. is east, 45*, 
30' 

6. 1st. from E., 
2nd. from W. 

7. 20*15'; 92* 

10. 64*33'; 54' 

11. 166°23'24'; 
llhrs. 6min., 
33.68ec. 

12. 3*46'46'; 
16min., 7Asec. 

13. 17**5r; Ihr.. 8 
min., 20sec. 

14. 2°,16',33.38ec.; 
9min., 2.228ec. 

Page 302 

17.1. 73*,44'59.85'' 

2. 3*.2',8.25'' 

3. 71°,7',45.76'' 

4. 77*,3',56.7'' 

18. Allegheny. 25 
min., 2.948ec. 
Berkeley, 2hrs., 
34min.. 7.638ec. 
Ann Arbor, 10 
hrs., 26min.t 
10.938ec. 
Chicago, Ihr., 
6min., 55.79sec. 
Chicago, 6hrs., 
59min., 47.81 sec 
Madison, 5hrs., 
44min.. 53.73 
sec. 

Cape G. Hope, 
3hr8., 37min., 
20.988ec. 
Paris, 4hr8., 41 
min., 54.778ec. 

19. 6* 15' 44.1" 
38* 31' 52.96'' 

156* 32' 43.96*' 
16* 28' 56.86* 
89* 66' 67.16* 
86* 13' 25.95* 
64* 20' 14.70' 
70* 28' 41.65" 



ANSWERS 



413 



Page 304 

1. 9 A.M; 8A.M.; 

8 A.M.; 8 A.M.; 
7 A.M.; 6 A.M.; 

6 A.M. 
a.New Or., 9 hr., 

10 min., 45 sec, 

P.M. 

PhUa.. 10 hr., 

10 min., 45 sec, 

P. M. 

Buffalo* either 

of the above 

Washington, 10 

hr., 10 min., 45 

sec, P. M. 

Austin, 9 hr., 10 

min., 45 sec, 

P.M. 

Boise City, 8 hr, 

10 min., 45 sec, 

P.M. 

Los Angeles, 7 

hr., 10 min., 45 

sec, P. M. 

El Paso, 7. 8, 9 

hra, 10 min.f 45 

sec, P. M. 

Salt Lake City, 

7 hrs., 10 min., 
45 sec. P. M. 

P.M. 

S.NewOr. 2:16 

Phila. 3:15 

Buffalo 2:15 

3:15 
Wash. 3:15 

Austin 2:16 
Boise City 1:16 
LosAng. 12:16 
El Paso 2:15 

1:15 
12:15 
S.L.City 12:15 
Dodge ('ity, 3.- 
26 A. M. 
Dodge City, 
3:26 A. M. 

A.M. 
New Or. 4:25 
Phila. 5:25 

Buffalo 4:25 

5:25 
Wash. 5:25 

Austin 4:25 

Boise City 3:25 
LosAng. 2:25 
El Paso 3:25 

2:25 

1:25 

S.L.City 2:25 

6. 12 o'c noon; 

1P.M., 4 P.M. 

9 P. M.; 11 P. 
M. 

6. 180° 

7. 1 hr. 

Page 806 

8. 12 o'c. noon; 
Sat. noon 



Page 306 

9.4 A.M. Wed.; 

6 A. M. Wed.; 

3 A.M. Wed.; 

2 A. M. Wed. 

lO.l. 8.25 P.M. 

2. 8.25 'P.M. 

3. 9.25 P.M. 

4. 10.25 P.M. 

5. 6.26 P. M. 

6. 8.26 P.M. 

7. 12.26 P. M. 

8. 11.25 P.M. 



Page 306 

2.81.60 per sq.ft.; 
814.40 per sq. 

yd. 
4. 17626 sq. ft.; 
16,800; 8000 
26,250; 5,200; 
26,312.5 



Page 307 

8. 90,000 cu. ft.; 

68,060; 75,000; 

90,000; 126.000 

10. 1. 13,333i sq. 

yds. 

2. 13,333i sq. 
yds. 

3. 10,000 sq. 
yds. 

4. 11,544^ sq. 
yds. 

5. 10,7361 sq. 
yds. 



Page 308 

12. 800; 889; 

1.333i; 1143 
13.4* to weather, 

838 . 72 

5" to weather, 

830.98 

Cost of tin88.16 
14. 31.445 

16. 8113.202 

16. 8220.11 

17. 18,191i 



Page 309 

19. 8445.536 

20. 8202.84 

21. 1728; 847.52 
23. 26.880; 8168 
84. 53,089i; 8742.- 

26 



Page 312 

2. 80; 80; 80; 40; 
38; 39; 40 

8.8. Perry, Lot 51 
No. A., 81 
J. Hay, Lot 50. 
No. A., 82 
J. Hay, E. i N. 
W4i. Sec. 7, No 
A. 80 

H. Ochilltree, 
lot 49, No. A., 
41 

H. Ochill. S. E. 
i, N. W. i. Sec. 
6, No. A. 40 
H. Ochill., E. i, 
S. W. i, Sec 6. 
No. A., 80 
J. Church, lot 
48. No. A. 38 
J. Church, lot 
47. No. A. 41 

Page 313 

6. 32 cu. in. 
6. little more than 

24 cu. in. 

7. 1. 76.3984 

2. 269.392 

3. 667.590 

4. 2150.4252 

5. 296.8812 

^•:7854-^- 

1416 ar2 
9. 761.73? gal. 
10. 9.76 min. 

11 14.72625 lbs. 

Page 314 

12. 211.58 

13. 562.699 

14. 4 
16. h 360°; 46° 
18.1. 16.6 

2. 188* 

3. 27.2485? 

4. 41.88628274 

Page 316 

19. 21.2142) 

20. 169. 7U 

21. 148.5 

22. 1.80796130? 

23. 27.2397321? 
34. 5903.9357142? 
26. 7.07J; 28.28| 

28. 1257* ; 314? 

29. 19704056 U sq. 
mi. 

Page 316 

30. S = 47rR» 

31. 120 cu. in. 
82. V-2ip 
33. 64 cu. in. 



Page 316 

34. 91,636,272 cu. 

ft. 
36. 7i?I?r< 

36. 2,234,281,760 
cu. ft. 

37. 12.330,981,000 
cu. ft. 



Page 317 

39. 33.51 + 

40. 33.51 + ; 2.807; 
4. 50+; 547.- 
7362911 

41.Circum. mi. 
Moon. 6,788^ 
Merc, 9,522? 
Venus, 24,300 
Earth. 24.886} 
Mars. 13.294? 
Jup.. 271,857^ 
Sat., 227.428^ 
Uran., 100,257f 
Nept., 109.371? 
Sun, 2.722,971? 
Surface 

Moon, 14.463,- 
314§ 

Merc, 28,854,- 
2671 

Venus, 186,340, 
000 

Earth, 197,040, 
66U 

Mars. 56,234,- 
828^ 

Jupiter, 23,515, 
642,8574 
Saturn, 16,748,- 
285,714? 
Uranus, 3,198,- 
202,8571 
Neptune, 3.806, 
125,714? 
Sun, 2.359,182,- 
446,714? 
Volume mi. 

Moon, 6,278,- 
793.142? 
Merc 14,671,- 
399,8541 
Venus, 239.136, 
333,333il 
Earth, 260.027, 
860,52Ut 
Mars, 39,645,- 
664,142? 
Jup., 339,017,- 
184,523,809U 
Sat., 203,770,- 
809,523.809^} 
Uranus. 17,003, 
778.523,809U 
Nept., 22,075.- 
529,142,867J 
Sun, 340,666,- 
946,161.142.- 
8571 



414 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



Pace 321 

1. 5 

2. 8 

3. 1. 15 

2. 16 

3. 20 

4. 36^ 
6. 28 
6. 30 

Page 324 

11. 1. 22 

2. 27 

3. 34 

4. 25 
6. 36 

6. 75 

7. 87 

8. 97 

9. 98 

10. 332 
14. 1. 43.6 

2. 68.7 

3. 71.1 

4. 7.82 

5. 8.71 

6. 8.94 

7. .74 

8. .677 

9. 1 . 103 

Page 326 

16. 1. 2.236 + 

2. 2.645 + 

3. 3.872 + 

4. .921 + 

5. .6 

6. 11.180 + 

7. 41.024 + 

8. 82.914 + 

9. 85.603 + 

10. 1.359 + 

11. 6.175 + 

12. 8.055 + 
16. 1. i 

ll if 

4. m 

5. AS 

6. i*flV* 

7. 



b* 



8. 



9. 85.603 + 

10. 1.359 + 

11. 5.175 + 

12. 8.055 + 

6. i"r 

6. /i 

7. A 

8. i? 



9. 



19. 
20. 



b 
42.8+* 
70.037+' 



Page 326 

21. 90' 

22. 4321.1 972 » 

23. 38.777 

24. 11248.72 sq.ft. 

Page 326 

26. 80' 

26. 133iyd8. 

27. 660 ft. 

28. 173.2 sq.ft. 

29. 3366.36 sq. ft. 

30. 804.60 sq. ft. 

1. 125 cu. in.; 
216 cu. in.; 343; 
1331; 5832; 9- 
261; 13.824 

Page 327 

2. 2 and 3; 3 and 
4; 4 and 5; 7 
and 8; 8 and 9; 
9 and 10; 38 and 
39; 78 and 79 

3. 15.625; 300.- 
763; 1953.125; 

37,037,V; 6591 

7.*J; I; ?; ,-o; fo; 



Page 330 

1. 62* 

2. 518 ft. 

3. yes 

4. 45 ft. 
6. 67* ft. 

6. 23f 

Page 331 

7. 128 rods 

9. r 

10. 3' 

11. 2200 mi. 

Page 332 

13. 704.545Vi miles 

14. 852.500 miles 
16. 1700 ft. 

Page 333 

16. 1504 ft. 

17. 54 ft. 

18. 846.4 

Page 336 

20. 4^; 4.52*; 
3.78'; 5.27*; 
5.35"; 8.32* 

21. 280'; 450'; 512' 
295': 625'; 448' 

22. 3.39*; 3.6*; 3.- 
425*; 3.69»; 3.- 
49*; 3.125'; 3.- 
395* 

23. 762'; 824'; 572' 
550'; 1172'; 12- 
36'; U96' 



Page 336 

26. 988 sq. ft. 

26. 607.3 sq. ft..; 
774; 917.2; 749 
1672.5; 756.3 

1. $11.90 

2. 36250 

3. 14.000 

4. 32.800 
6. $16,000 



Page 337 




6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
16. 
16. 
17. 
18. 
19. 



20. 76%; 667.60 



Page 338 

1. $24087. 6Q 

2. 33.874. 36i 

3. $7390.512 

4. $684,042.55 
6. $3312133i 

6. 18 mills on a 
dollar 

7. $3001.471 

8. 15 

9. $158.40 

10. $21,443 

11. $84 

Page 339 

12. $100 

13. .013;$13.639.11 

14. $12767.48 
16. 19 
16. $21,354.16 

1. $578.86; 
$672.76 

Page 340 

2. $143.66 

3. $145.19 

4. $1051.84; 
6. $1040.77 
6. $286,226 

Page 342 

1. 2.1%; $420 

2. $26350; $26550 
$26,200; $26400 

3. $35,000 
$36,060; $34.- 
950; $34,960 

4. Gain, $600 
6. Neither loses 

nor gains 

6. $1662.60 

7. M 

8. 9.7% 



Page 342 

9. 1.85%; 1.83% 

10.1.2.76%; 2.74% 

2.2.76%; 2.74% 

3.3.55%; 3.53% 

4.2.80%; 2.85% 

11. 3.69% 

Page 343 

12. 4.66%; 4.55% 
13.Wab. R. R., 6s. 

.42%; .41% 
14. Strgt. loan. 

.06% 
16. 14.92% ;1 122% 

$8860 

Page 344 

2. $60; $7.68 

8.^ $112.61 + 
9. $277.46 

Page 346 

10. $226.67 

11. $631.24 

12. $614.43 

2. x + y 

3. x + y 

4. 80c. -45c. «= 
36c. 



6. 


m — 8 dollars 


6. 


45c. 


7. 


zy dollars 




Page 346 


8. 


5A° 


9. 


7 rods 


10. 






y 


11. 


SO'-.; 40c. 


12. 


13. 


8x; 3x' 


16. 


28'* 


16. 


12 + x; 12-y 


17. 


2**. 6*», 12« 


18.31.50 hat; $3.00 


( 


:oat 


19. 


x-$30 




x-30 


20. 


2 




Page 347 


21. 


x-y 


22. 


3a 


23. 


x-y 


24. 


a + b 


26. 


3 + a miles per 




hr. 


26. 


3-a 


27. 


5a cents 


28. 


ax c. 


29. 


$40 


80. 


15 




X 


31. 


-S-rd.. 


32. 


X-30 



ANSWERS 



415 



34. 



c. 



33. lOx +8yc 

m + c 
3 
3R. 5 

+ c) c. =(m + 
2c) c 
36. 72 xy. sq. in. 



(=1^") 



Pase 348 



37. 
38 



3 



bx + by or 
6(x + y) 
1. 75 lbs 

3. p = 75 lbs. 

3. F = 300 lbs. 

4. F = 7956 lbs. 

5. F» 3978 lbs. 



Page 349 

3. 2 

4. 10; 7 

6. b; (b + c) lb.; 
(p + 36) lbs. 

Page 362 

1. 1. 61bs. 

2. 3 ft. 

3. 2^ mi. 

4. 4c. 

5. 14 

6. 21 

7. 6 

8. 24 

9. 3 

2. 6; 21; U 

3. 15; 2; 19 

Page 363 

4. 5; 15; 25; 25 
6. 4;12; 28; 16; 2 

6. 9; 27; 81; 3 

7. 3 
8 21i 
9. 4 

10. 2:2 

11. 7;7 
12.10; 5; 3i; 2i; 

1; i 

14. 1. 13 

2. 3 

3. 80 

4. 1600 

5. 9 

6. 89 

7. 9 

8. 1600 

9. 39 

10. 89 

11. 712 

12. 15 



Page 363 

16.1. x<y correct 

2. x<y 

3. x<y 

4.y> —X correct 
5.x + y=13 cor. 
6.x-y = 6 cor. 

7. (x + y)2 = 
1296 

8. (x-y)2 = 25 
cor. 

9. x2-y2<25 
cor. 

10.x2 + y2=97 
ll.(x + y)» = 
x^ + 2xy + y2 
12.(x-y)2 = 
x'-*— 2xy + y2 
13.(x + y)2 = 
x2 + 2xy + y2cor 
14.(x-y)2 = 
x2-2xy + y2cor 
15.(x-y)2 = 
x2-2xy + y2 
16.(x + y)(x- 
y) = x«-y^ 
17.(x + y)(x- 
y) = x*-y2 
18.(x-y) (X- 
y) = x2-y2 
I9.{x-y)ix + 
y) = x2-y2 
20.x(x + y) • = 
x^ + xy cor. 



Page 366 

6. 1. 
2. 
3. 
4. 
5. 
6. 
7. 



2 
2 
3 
5 
4 
2 
614 

Page 367 

4. 4 no. ash trees 
12 no. oak trees 
16 no. hickory 
trees 

6. 15 5c. stamps 
60 2 c. stamps 
30 10 c. stamps 

6. 20 acres 

7. 6; 24; 3 

Page 368 

8. 9; 18; 21 

9. 56,650 sq. mi. 

10. 16 c. 

11. 126f mi 

12. 285 mi. 

13. 58 mi. 

14. 248 mi. 
16. 369 mi. 

16. Son. 315; 
Father, $60 

17. Joseph 6; 
William 18 

18. 12; 20 

19. 30; 45 

20. 40 



Page 369 

21. 440 rods 

22. $2.50 

23. 40 

24. $37.50 
26. 350 rods 
26. 8; cir. »50| 
27.R= y"^; C 

= 6? ^IT 



1. 

2. 

3. 

4. 

6. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
16. 
16. 
17. 
18. 
19. 
20. 



3 

8 

7 

2 

24 

36 

20 

12 

3 

10 

3 

2 

2i 

8 

15 

12 

8 

9 

3 

3 



Page 360 

21. Hemp cable 
1" diam. 

W. .577 sq. in. 

S. .109 sq. in. 

S. .654 sq. in. 

Hemp cable 2^" 

diam. 

W. 3.60625 sq. 

in. 

L. .68125 sq. in. 

S. 4.0875 sq. in. 

22. Tarred hemp 
rope, 1* diam. 
W. 1.036 sq. in. 
L. .247 sq. in. 
S. 1.480 sq. in. 
Tarred hemp 
2^'* diam. 

W. 6.475 sq .in. 

L. 1.54375 sq. 

in. 

S. 9.25 sq. in. 

Manila rope I'' 

diam. 

W. .765 sq. in. 

L. .329 sq. in. 

S. 1.877 sq. in. 

Manila rope 2i* 

diam. 

W. 4.78125 sq. 

in. 

L. 2.05625 sq. 

in. 

S. 11.73125 sq. 

in. 



Page 360 

23. Iron wire li* 
diam. 

W. 6.0109375 

sq. in. 

L. 4.471875 sq. 

in. 

B. 26.58125 sq. 

in. 

Iron wire, 24* 

diam. 

W. 17.371609- 

375 sq. in. 

L. 12.92371875 

sq. in. 

S. 76.8198125 

sq. in. 

24. Steel wire, li" 
diam. 

W. 6.165625 sq. 

in. 

L. 6.9390625 sq. 

in. 

S. 43.171875 sq. 

in. 

Steel wire, 24" 

diam. 

W. 17.81865626 

sq. in. 

L. 20.053890625 

sq. in. 

S. 124.76671876 

sq. in. 

Page 361 

2. 12; 6 

3. 

4. 

6. 26 girls;: 

6. 125; 50 

7. 24; 16 

8. 15i; 7i 

9. 12i; 7i 

11. 7; 21 

12. 371.25; 41.25 

Page 362 

13. 3 tin; 12 copper 

14. 224tons copper 
2i tons tin. 

16. 2034; 28 leu. in. 

16. 7; 2 

17. 8; 5 

18. 3; 7 

19. 15 in.; 9 in. 

20. 24; 8 

21. 16; 12 

22. 24; 40 

23. I 
26. 2; 12 

Page 363 

26. 2i; 3^ 

27. -10; -35 

28. 2; 3 

29. x=12; y = 5 

30. x = 14 

y»3 

31. x = 10 

y = 6 



if'^ 



416 



RATIONAL GRAMMAR SCHOOL ARITHMETIC 



82. 
S3. 
84. 
85. 
86. 



y = 5 

y-14 
x-1 



ii 



b=-2H 
cl«2f 

Pace 864 

1.212°F; 100**C.; 

SO'^R. 
2.32°F.; 0**C;0**R. 
8. 80*»R.; 100°C.; 

180°F. 

Page 867 

l.Ice, 32*»F.; 0*»R. 
Benaol, 39.9°F.; 
3.52«R. 

Tallow, 109.4° 
F.; 34.4*» R. 
Parraffin, 114.- 
8**F.; 36.8*» R. 
Wax. 143.6° F; 
49.6* R. 
Sulphur. 239° 
F.; 92°R. 
Tin, 446°F.; 
184°R. 

Bismuth. 482° 
F.; 200°R. 
Cadmium. 608° 
F.; 256°R. 
Lead, 618.8° F.; 
260.8°R. 



Page 867 

Zinc, 773.6°F.; 
329. 6°R. 
Antimony, 829- 
.6°F.; 346.6°R. 
Silver, 1832°F.; 
800°R. 

Copper, 2012° 
F.; 880°R. 
Gold, 2192°F.; 
960°R. 

Caflt-lron, 2192° 
F.; 960°R. 

Cast-Steel, 
2507°F.; 1100° 
R. 

Wrought-Iron, 
291 2°F.; 1280° 
R. 

Platinum, 3227° 
F.; 1420°R. 
Iridium, 
3642°F.; 1560° 
R. 

Page 868 

2. Ether, 35°C.; 
28°R. 

Carbon Dis., 
46°C.; 36.8° R. 
Sulph. Acid, 
-10°C.; -8.° 
R. 

Chloroform, 
61°C.; 48.8°R. 
Alcohol, 78°C.; 
62.4°R. 



Page 868 

Water, 100°C.; 

80. °R. 

Mercury, 357° 

C; 285.6°R. 

Zinc, 1040°C.; 

832°R. 
4. 98.96° Fahr. 
29.76° Ream. 
6.Ether vapor, 

384.8° Fahr. 

Carbonic acid, 

87.8° Fahr. 

EtheUne, 48.2° 

Fahr. 

Oxygen, 180.4° 

Fahr. 

Nitrocen, 229° 

Fahr. 

Hydrogen, 

281.2° Fahr, 

Page 870 

8. 8114.5625 

8. 8338.125 

4. 5.5; 2.40625 

sq. in. 
6. 254.57 

6. 48.43"^ 

Page 871 

2. 8156.45 

Page 272 

2. 1. 884 

2 396 

3. 1591 

4. 2436 

5. 3675 



Page 872 

6. 8036 

7. 4899 

8. 8019 

9. 8075 
8. 1. 1296 

2. 2209 

3. 4096 

4. 3364 

5. 361 

6. 9025 

7. 15625 

8. 21904 

Page 878 

6.1. 114.43 

2. 20159.81 

3. 8.56 

4. 4780.30 
7. 1. 248.729 

2. 496.885 

3. 1977.279 

4. 4035.960 
2. 1. 568.75 

2. 73.27 

3. 42.51 

4. 2.39 
Page 874 

2. 1. 80.20 

2. 196.54 

3. 93.62 
4. 1. 34 

2. 212 

3. 13.2 

4. 222 

5. 234 

6. 638